FIXED POINTS OF RENORMALIZATION.

Size: px
Start display at page:

Download "FIXED POINTS OF RENORMALIZATION."

Transcription

1 FIXED POINTS OF RENORMALIZATION. XAVIER BUFF Abstract. To study the geometry o a Fibonacci map o even degree l 4, Lyubich [Ly2] deined a notion o generalized renormalization, so that is renormalizable ininitely many times. Van Strien and Nowicki [SN] proved that the generalized renormalizations R n () converge to a cycle { 1, 2 } o order 2 depending only on l. We will explicitly relate 1 and 2 and show the convergence in shape o Fibonacci puzzle pieces to the Julia set o an appropriate polynomial-like map. Keywords. Holomorphic dynamics, renormalization, Fibonacci. 1. Introduction. In this article, our goal is to study the geometry o real Fibonacci maps o degree l 4. The importance o Fibonacci maps has been emphasized by Hobauer and Keller [HK] or unimodal maps and by Branner and Hubbard [BH] or cubic polynomials. In [LM], Lyubich and Milnor studied the restriction to the real axis o a quadratic Fibonacci polynomial, and this study was enlarged to the complex plane by Lyubich in [Ly2] and [Ly3]. Existence o real Fibonacci polynomials o the orm z z l + c was obtained by Hobauer and Keller or any even integer l 2, and ollows rom a combinatorial argument due to Milnor and Thurston [MT]. However, Lyubich and Milnor [LM] observed that the geometry o Fibonacci maps was dierent or degree l = 2 and or degrees l 4. Fibonacci maps o degree l 4 have since been studied by van Strien and Nowicki in [SN] where they obtained new results using renormalization techniques. We would like to use results by H. Epstein [E1] [E2] on ixed points o renormalization to improve the results obtained by van Strien and Nowicki. In his survey [Ly4], Lyubich describes renormalization in the ollowing way: the notion o renormalization o a dynamical system consists in taking a small piece o the dynamical space, considering the irst return map to this piece, and then rescale it to the original size. The new dynamical system is called the renormalization R() o the original one. Depending on the way one chooses the small piece, and the way one deines the irst return map, one gets several deinitions o renormalization. We will show that the study o the geometry o real Fibonacci maps o degree l 4 is similar to the study o the geometry o Feigenbaum maps. For this purpose, we will show that one can make a parallel approach between two notions o renormalization that have been developped during the last two decades in holomorphic dynamics. The irst notion o renormalization was introduced in 1976 by Feigenbaum [F1] [F2], and independently Coullet & Tresser [CT] or real dynamical systems and more precisely or unimodal maps. To explain a universality phenomenon, they deined a renormalization operator R that acts on an appropriate space o dynamical systems, and conjectured that R had a unique ixed point. Lanord [La] gave a 1

2 2 X. BUFF computer-assisted proo o this conjecture. Later, Epstein [E1] [E2] gave a proo o the existence o a renormalization ixed point that does not require computers. However, his proo does not give uniqueness o the ixed point. This ixed point satisies a unctional equation known as the Cvitanović-Feigenbaum equation: (z) = 1 α (αz), or some α ]0, 1[. In 1985, the generalization to holomorphic dynamics via polynomiallike mappings, was introduced by Douady and Hubbard [DH]. The classical renormalization theory has been extensively studied (see Collet and Eckmann and Lanord [CE], [CEL] and [La], Cvitanović [Cv], Eckmann and Wittwer [EW], Vul, Sinai and Khanin [VSK], Epstein [E1] and [E2], Sullivan [S], de Melo and van Strien [dmvs], McMullen [McM1] and [McM2], Lyubich [Ly3], [Ly4], [Ly5] and [Ly6]). For a historical account, the reader is invited to consult [T] or [Ly5]. Lyubich generalized the notion o renormalization or polynomial-like mappings, to a wider class o maps, that we will call L-maps. This allowed him to apply the renormalization ideas to non-renormalizable maps as well. Lyubich and Milnor [LM] showed that this generalization could be applied to the study o Fibonacci maps. Let us deine a Fibonnaci map as a branch covering : U 0 U 1 V, such that U 0, U 1 and V are topological open disks satisying U i V, i = 1, 2 and U 0 U 1 = ; has a unique critical point ω U 0 : the orbit o the critical point satisies some combinatorics that will be deined in section 4. Fibonacci maps are not renormalizable in the classical sense. However, it has been the idea o Lyubich that one could deine a generalized renormalization operator R, sending the space o Fibonacci maps into itsel. Hence, given a Fibonacci map, one can deine an ininite sequence o generalized renormalizations R n (). In [SN], van Strien and Nowicki proved that i the degree l o the critical point ω is larger than 2, and i the map is real (i.e., ( z) = (z)), then this sequence converge to a cycle { 1, 2 } o order 2, where 1 and 2 are two Fibonacci maps o degree l. In [Ly4], Lyubich writes: the combinatorial dierence between 1 and 2 is that the restrictions o these maps on the corresponding non-critical puzzle pieces have opposite orientation. We will prove that in act 1 and 2 are related in the ollowing way. Theorem A. For every even integer l 4, let i : Ui 0 U i 1 V i, i = 1, 2, be the two real Fibonacci maps o degree l, normalized so that ω i = 0 and i (ω i ) = 1, and satisying R( 1 ) = 2 and R( 2 ) = 1. Then, there exists a neighborhood U o 0 and a neighborhood U o 1 such that 1 U U 0 1 = 2 U U 0 2, and 1 U U1 1 = 2 U U2 1. The main ingredient in our proo is a lipping operator that does not preserve the dynamics o the maps, but has the nice property o sending the space o real Fibonacci maps o degree l into itsel.

3 FIXED POINTS OF RENORMALIZATION. 3 W α W α 2 1/α 2 J( α ) x 0 /α 1/α 2 x 0 /α 2 J( α 2) α α 2 Figure 1. The polynomial-like maps α : W α α (W α ) and α 2 : W α 2 α 2(W α 2) corresponding to a degree l = 6 and their Julia sets J( α ) and J( α 2). We will then show that the restriction o 1 to U 0 1 satisies a system o equations, that we will call the Cvitanović-Fibonacci equation: (z) = 1/α 2 (α(αz)), 0 < α < 1, (0) = 1 and (z) = F (z l ), with F (0) 0 and l 4 even. We will irst study the geometry o the solutions o the Cvitanović-Fibonacci equation, and we will prove the ollowing theorem. Theorem B. For every even integer l 4, let be the solution o the Cvitanović- Fibonacci equation in degree l, and set α (z) = (αz) and α 2(z) = (α 2 z). Then, there exist domains W α C and W α 2 C containing 0 such that α : W α α (W α ) and α 2 : W α 2 α 2(W α 2) are polynomial-like mappings o degree l. Besides, α : W α α (W α ) has an attracting cycle o order 2 and α 2 : W α 2 α 2(W α 2) has an attracting ixed point. In particular, the Julia set J( α ) is quasi-conormally homeomorphic to the Julia set J(z z l 1) and the Julia set J( α 2) is a quasi-circle. Finally, the domain o analyticity o is the quasi-disk Ŵ bounded by the quasicircle αj( α 2). Figure 1 shows the two polynomial-like mappings α : W α α (W α ) and α 2 : W α 2 α 2(W α 2) and their Julia sets. Remark. In the context o classical renormalization, McMullen proved that the domain o analyticity o the ixed point o renormalization satisying the Cvitanović- Feigenbaum equation is a dense open subset o C. Our result shows that the behaviour or generalized renormalization is drastically dierent.

4 4 X. BUFF Figure 2. Degree six Fibonacci puzzle pieces (made by Scott Sutherland). The next step will be to prove that any solution o the Cvitanović-Fibonacci equation gives rise to a cycle o order 2 o Fibonacci maps which is invariant under renormalization. Theorem C. Given any solution o the Cvitanović-Fibonacci equation, there exists a Fibonacci map φ : U 0 U 1 V such that φ and coincide on U 0 and such that R 2 ([φ]) = [φ]. We will then derive the ollowing corollary. Corollary. For every even integer l 4, there exists a unique α ]0, 1[ such that the Cvitanović-Fibonacci equation has a solution, and this solution is itsel unique. We will say that is the solution o the Cvitanović-Fibonacci equation in degree l. We will then deine a Yoccoz puzzle or Fibonacci maps, and study the convergence in shape o puzzle pieces. In [Ly4], Lyubich writes: the ollowing picture o the principal nest or degree 6 Fibonacci map show that all puzzle pieces have approximately the same shape: [see igure 2] [... ] these puzzle pieces have asymptotically shapes o the Julia set o an appropriate polynomial-like map. We will prove that this observation is true. More precisely, we will prove the ollowing theorem. Theorem D. Let S k be the Fibonacci numbers deined by S 0 = 1, S 1 = 2, and S k+1 = S k + S k 1, l 4 be an even integer, F : U 0 U 1 V be a real Fibonacci map o degree l normalized so that the critical point is ω = 0,

5 FIXED POINTS OF RENORMALIZATION. 5 C k be the connected component o F k (V ) that contains the critical point (it is called the critical puzzle piece o depth k), be the solution o the Cvitanović-Fibonacci equation in degree l, α ]0, 1[ be the constant deined by the Cvitanović-Fibonacci equation, and α : W α α (W α ) and α 2 : W α 2 α 2(W α 2) be the polynomial-like mappings deined in Theorem B. Then, there exists a constant λ 0 such that the sequence o rescaled puzzle pieces λ α k+1 C S k 2 converges or the Hausdor topology to the illed-in Julia set K( α 2), and λ the sequence o rescaled puzzle pieces α k 1 C S k 3 converges to the illed-in Julia set K( α ). Let us mention that a similar result has already been proved by Lyubich [Ly2] or Fibonacci maps in degree 2. He proved the convergence in shape o some puzzle pieces to the Julia set o z z Dynamical systems. In this section, we will quickly recall the deinition o polynomial-like mappings (see [DH]) and o generalized polynomial-like mappings (see [Ly3]). We will also deine the corresponding notion o renormalization Polynomial-like maps. In [DH], Douady and Hubbard introduced the concept o polynomial-like maps. A polynomial-like map is a branched covering : U V between two topological disks U and V, with U V. One deines the illed-in Julia set K() and the Julia set J() o a polynomial-like map : U V as: K() = {z U ( n N) n (z) U}, and J() = K(). Deinition 1. We say that is a DH-map i : U V is a polynomial-like map having a single critical point ω K(). Remark. The Julia sets K() and J() are connected i and only i K() contains all the critical points o. Hence, the Julia set o a DH-map will always be connected. Douady and Hubbard showed that a polynomial-like map behaves dynamically like a polynomial. Proposition 1. (see [DH]) For each DH-map, there exists a unique polynomial P c o the orm z l +c, up to conjugacy by z e 2iπk/(l 1) z, k = 0, 1,..., l 1, topological disks U c and V c, and a quasi-conormal homeomorphism φ : V V c satisying φ/ z = 0 a.e. on K(), such that or all z U, φ = P c φ. We will say that the two maps are in the same hybrid class. Deinition 2. Given a DH-map : U V, we say is renormalizable i we can ind an integer k strictly greater than 1, and

6 6 X. BUFF topological disks U 1 and V 1 containing ω, such that k : U 1 V 1 is a DH-map. We will say is k-renormalizable and that k : U 1 V 1 is a renormalization o. We will be interested in one particular polynomial, called the Feigenbaum polynomial, which is the unique polynomial, P F eig = z 2 + c F eig with c F eig R, 2-renormalizable, and such that P F eig and its renormalizations are in the same hybrid class. Results about the existence and uniqueness o this polynomial are discussed in [S], in [dmvs] and in [McM2] Generalized polynomial-like maps. Another amily o polynomial-like maps was introduced by Lyubich in [Ly3] to study what happens when the critical point escapes rom U. This kind o maps appears naturally when one studies cubic polynomials with two critical points, one escaping to ininity, the other having a bounded orbit (see [BH]). Deinition 3. A generalized polynomial-like map, which we will call an L-map, is a ramiied covering map : k 1 i=0 U i V such that there is a unique critical point ω U 0, the orbit o ω is contained in the union o the U i, each U i contains at least one point o the orbit o ω, and i i < j the orbit o ω visits U i beore U j, each U i, i = 0,... k 1 is a topological disk compactly contained in V, the U i are pairwised disjoint. There is only one way o ordering the U i because o the third condition. We can again deine K = {z n (z) k 1 i=0 U i, n N}, and J() = K(). Figure 3 shows an L-map : U 0 U 1 V, where is a cubic polynomial with one critical orbit escaping to ininity and one critical point having a bounded orbit. As in [BH], [H] or [Mi], we can deine the puzzles associated to an L-map. Deinition 4. The puzzles are deined by induction: the elements o the puzzle P 0 () o depth 0 are the open sets U i (called puzzle pieces), the elements o the puzzle P n o depth n are the connected components o n (P 0 ). We can deine a notion o renormalization associated to those L-maps. Deinition 5. We will say is L-renormalizable i we can ind a inite collection o puzzle pieces U i 1, i = 0,..., l 1, a puzzle piece V 1 and integers n i, i = 0,..., l 1 with at least one n i > 1, such that ω U 0 1 and

7 FIXED POINTS OF RENORMALIZATION. 7 V U 1 U 0 Figure 3. the L-map : U 0 U 1 V and its Julia set. l 1 the map g : i=0 U 1 i V 1 deined by g U1 i = n i is an L-map. l 1 We say is (n 0,..., n l 1 )-renormalizable, and g : i=0 U 1 i V 1 is an L- renormalization o. In this context, we will study Fibonacci maps, which are L-maps deined by some dynamical properties. Such maps were introduced in [HK] and [BH], and studied urther in [LM], [Ly2] or [SN]. For the two kind o renormalizations, the maps we are interested in are ininitely renormalizable. We will assume the sequence o successive renormalizations converges to a ixed point o renormalization (c [S], [dmvs] or [McM2] or Feigenbaum case, and [SN] or Fibonacci maps. We will then study those ixed points o renormalization, using H. Epstein s work (c [E2]). To do this, we must irst introduce two notions. The irst one is a notion o convergence, which will enable us to talk o limits, the second one is a notion o germs which will allow us to talk o ixed points o renormalization Topology on the space o polynomial-like maps. In [McM1], McMullen introduces the ollowing topology. First o all, a pointed region is a pair (U, u), where U C is an open set, and u U is a point. Deinition 6. We say that (U n, u n ) converges to (U, u) in the Carathéodory topology i and only i u n u, and or any Hausdor limit K o the sequence P 1 \ U n, U is the connected component o P 1 \ K which contains u. To deine a topology on the sets o polynomial-like maps, we use a theorem by McMullen.

8 8 X. BUFF Proposition 2. (see [McM2]) Let g n : (U n, u n ) (V n, v n ) be a sequence o proper maps between pointed disks, with deg(g n ) d. Suppose u n u, g n converges uniormly to a non-constant limit on a neighborhood o u, and (V n, v n ) (V, v). Then (U n, u n ) converges to a pointed disk (U, u), and g n converges uniormly on compact subsets o U to a proper map g : (U, u) (V, v), with 1 deg(g) d. This enables us to deine a topology on the sets o DH-maps or L-maps, because each branch o those maps are proper maps between disks. For DH-map and L- map, there is a natural way o choosing the basepoints. One can, or example, take the irst visit o the critical orbit in the disks U i Space o germs. In [McM1], McMullen introduces the notion o germs o polynomial-like maps. We can adapt this notion to L-maps. To do so, we just need to say two maps 1 and 2 are equivalent i they have the same Julia set, K( 1 ) = K( 2 ) = K, and i 1 K = 2 K. Deinition 7. The set G o germs [] is the set o equivalence classes. McMullen gives G the ollowing topology: [ n ] [] i and only i there are representatives n and, which are DH-maps or L-maps, depending on the context, and such that n or the Carathéodory topology. Then, the space o germs is Hausdor. 3. Feigenbaum maps All the results we will state here have already been proved by Epstein [E1] and McMullen [McM1]. The goal is to introduce some unctional equation satisied by ixed points o renormalization, and to state some results related to it. The work we present here has been completed in [B3] Feigenbaum polynomial. The Feigenbaum polynomial is the most amous example o polynomial which is ininitely renormalizable (meaning it is k-renormalizable or ininitely many k). It is the unique real quadratic polynomial which is 2 k - renormalizable or all k 1. Deinition 8. One can deine the Feigenbaum polynomial as the unique real polynomial which is a ixed point o tuning by 1. Tuning is the inverse o renormalization. Given a parameter c M, such that 0 is a periodic point o period p, Douady and Hubbard have constructed a tuning map, x c x, which is a homeomorphism o M into itsel, sending 0 to c, and such that i x 1/4, then c x is p-renormalizable, and the corresponding renormalization is in the same hybrid class as x. This is how they show there are small copies o the Mandelbrot set inside itsel. The Feigenbaum value, c F eig = , is in the intersection o all the copies o M obtained by tuning by 1. This intersection is not known to be reduced to one point, but its intersection with the real axis is reduced to the point c F eig.

9 FIXED POINTS OF RENORMALIZATION Feigenbaum renormalizations. By construction, the Feigenbaum polynomial, P F eig, is 2-renormalizable. There are several renormalizations g : U V such that g = 2 U is a DH-map. But all those renormalizations deine the same germ o DH-map. By the straightening theorem (see [DH]), there is a unique polynomial which is in the same hybrid class as g, i.e., quasi-conormally conjugate to g on a neighborhood o its Julia set, the derivative o the conjugacy vanishing almost everywhere on the Julia set. This polynomial is a real polynomial, because g is real, and 2 k -renormalizable or all k 1. Thus, it is the Feigenbaum polynomial. We can deine a renormalization operator, R 2. Given a germ o DH-map, [], which hybrid class is the one o the Feigenbaum polynomial, let us choose a representative g corresponding to period 2 renormalization. Deinition 9. Assume [] is a germ o a quadratic-like map which is renormalizable with period 2. There exist open sets U and U such that the map g : U U deined by g = 2 U is a polynomial-like map with connected Julia set. The renormalization operator R 2 is deined by with α = g(0) = 2 (0), and α(z) = αz. R 2 ([]) = [α 1 g α], We have normalized the germs, so that the critical value is 1. We have seen that i [] is a germ o Feigenbaum DH-map, then R 2 ([]) is still a germ o Feigenbaum DH-map, and we can iterate this process, deining in such a way a sequence o germs : R n 2 ([P F eig ]), n N. The ollowing result can be ound in [S], [dmvs] or [McM2]. Proposition 3. The sequence o germs R n 2 ([P F eig ]), n N, converges (or the Carathéodory topology deined in the introduction), to a point [φ]. By construction this point is a ixed point o renormalization : R 2 ([φ]) = [φ], and is in the hybrid class o the Feigenbaum polynomial. It is the unique ixed point o R 2. Remark. We say that two quadratic-like germs [] and [g] are in the same hybrid class i there exist representatives : U U and g : V V which are in the same hybrid class. Now, i [] is a ixed point o R 2, then it satisies the ollowing unctional equation, known as the Cvitanović-Feigenbaum equation. Proposition 4. Let [] be a ixed point o R 2. Then, (z) = 1 α((αz)), 0 < α < 1, (0) = 1, and (z) = F (z 2 ), with F (0) 0. This equation is satisied at least on the Julia set o (which does not depend on the representative o the germ []) Study o some unctional equations. Now the question is: what inormation can we obtain rom this equation? The way we can deal with it, was explained to us by H. Epstein and is developped in [E2]. There is a global theory which enables us to deal with the study o the ixed points o the three kind

10 10 X. BUFF o renormalization at the same time. The unctional equation we will study will depend on two parameters. The irst one is the degree l o the critical point. The second one is a parameter ν which corresponds to the case we are dealing with. in the case o renormalization or DH-maps, ν = 1, in the case o renormalization or L-maps, ν = 1/2, and in the case o renormalization or holomorphic pairs, ν = 2. Deinition 10. The universal equation is the ollowing system o equations: (z) = 1 λ ((λν z)), 0 < λ < 1, (0) = 1, and (z) = F (z l ), with F (0) 0. First o all, we want to study solutions such that and F are real analytic maps on an open interval J containing 0, and their complex extension. So let J be an open interval in R, possibly empty, and deine where C(J) = {z C : Im(z) 0, or z J} = H + H J, H + = {z C : Im(z) > 0} = H. F(J) is the space o holomorphic unctions h in C(J), such that h(z) = h(z). P 1 (J) F(J) is the space o unctions h such that h(h + ) H +. A unction h P 1 (J) is called a Herglotz unction (and h is anti-herglotz). We will study solutions o the universal equation, such that F is univalent in a neighborhood o 0, and has an anti-herglotz inverse, F 1. In act, as the limit o renormalization can be obtained as a limit o polynomials having all their critical values in R, this condition is satisied by the ixed points o renormalization we will consider. The irst step is to look at the graph o on the real axis. Figure 4 shows what this graph looks like. This graph gives the relative position o several points on the real axis. Proposition 5. Epstein (see [E2]) Let be a solution o the universal equation, and x 0 > 0 be the irst positive preimage o 0 by. Then (λ ν x 0 ) = x 0, (1) = λ, and the irst critical point in R + is x 0 /λ ν, with (x 0 /λ ν ) = 1/λ. Besides, the universal equation can be restated in two surprising ways on the ollowing commutative diagrams. The irst diagram tells that x 0 is an attracting ixed point o the map (λ ν z). The linearizer is. x 0 (λ ν z) x 0 0 λz 0

11 FIXED POINTS OF RENORMALIZATION x 0 0 x 0 x 0 /λν λ ν x 0 1 λ 1/λ Figure 4. The graph o on R +. The second diagram tells that 1 is a repulsive ixed point o the map 1/λ(z). The parametrizer is F. F 1 0 1/λ(z) z/λ ν The irst diagram enables Epstein to study how much one can extend F 1. His results are the ollowing. Theorem 1. Epstein (see [E2]) Let (z) = F (z l ) be a solution o the universal equation, such that F 1 is anti-herglotz. We then have the ollowing results: one can extend F 1 such that F 1 P 1 (] 1/λ, 1/λ 2 [), one can extend F 1 continuously to the boundary R o H +, and even analytically except at points ( 1/λ) n, n 1, which are branching points o type z 1/l, the values o F 1 are never real except in [ 1/λ, 1/λ 2 ], the extension o F 1 to the closure o H + is injective, and when z tends to ininity in H +, F 1 (z) tends to a point in H, which will be denoted by F 1 (i ). By symmetry, similar statements hold in H. Hence, W = F 1 (C(] 1/λ, 1/λ 2 [) is a bounded domain o C. Those results are summarized in igure 5. In the ollowing, we will use the notations: By construction, C λ = C \ ( ], 1/λ] [1/λ 2, + [ ), W = F 1 (C λ ), and W = {z C z l W}. : W C λ 1 0 F

12 12 X. BUFF F 1 ( i ) W 0 R F 1/λ 1 1/λ 2 Figure 5. Maximal univalent extension o F. is a ramiied covering with only one critical point in 0, o degree l. The set W is symmetric by rotation o angle 2iπ/l. In [E2], Epstein uses those inormations to prove the ollowing result. Theorem 2. (c [E2] and igure 6) Let be a solution o the universal equation o parameters ν = 1 and l = 2. The map : W C λ is a polynomial-like map. It is quasi-conormally conjugated to the Feigenbaum polynomial P F eig. 1/λ K() W 1/λ 2 C λ Figure 6. The Feigenbaum map : W C λ. Proo. See [E2] or [B3]. To study the geometry o the Julia set o the Feigenbaum polynomial, it is sometime enough to study the geometry o the Julia set K() o this polynomiallike map. This has been done in [B2]. Some results are easier to obtain using the ixed point o renormalization because o the invariance with respect to the scaling map z λz.

13 FIXED POINTS OF RENORMALIZATION Fibonacci maps. In this section, we will deal with renormalization or L-maps, and more precisely, or Fibonacci maps. We will see that the dynamics o Fibonacci maps is strongly related to Fibonacci numbers, so let us irst recall the deinition o the Fibonacci sequence S n. Deinition 11. The Fibonacci sequence S k is deined by S 0 = 1, S 1 = 2, and S k+1 = S k + S k 1. In particular S 2 = 3, S 3 = 5, S 4 = 8, S 5 = 13, and so on Deinition o Fibonacci maps. Let us now return to the deinition o Fibonacci maps. We have deined the puzzles P n associated to an L-map in the introduction. Following Branner and Hubbard, we will distinguish the puzzle pieces that contain the critical point and deine a notion o genealogy between those pieces. Deinition 12. Let : U i V be an L-map, and or each z K(), let P n (z) be the puzzle piece o depth n which contains z. The critical piece C n o depth n is deined to be P n (ω) i n 0 and C 1 is deined to be the piece V. For all n 1, the children o C n are the critical pieces C l such that (l n) (C l ) = C n and k (C l ) does not contain the critical point ω or 0 < k < l n. Remark. I C l is a child o C n, then (l n) : ramiied only at ω. Let us now deine what is a Fibonacci map. C l C n is a ramiied covering Deinition 13. A Fibonacci map o degree l 2 is an L-map : U 0 U 1 V having a critical point o degree l and satisying the ollowing conditions: or each n 1, the critical piece C n has exactly two children; i C l is a child o C n, then (l n) (0) C n \ C l. Figure 7 shows the domain, range and Julia set o a Fibonacci map having a critical point ω o degree 6. We would like to mention that our deinition o Fibonacci maps is not the one given by Branner and Hubbard in [BH] but we will show that it is equivalent. The irst condition is not suicient to guaranty that those maps are Fibonacci maps in the sense o [BH]. The second condition says that there are no central returns in the terminology o Lyubich [Ly3]. The way Branner and Hubbard deine Fibonacci maps is the ollowing. They introduce the concept o a tableau in order to describe recurrence o critical orbits or cubic polynomials having one escaping critical point, and one critical point with bounded orbit. They call a Fibonacci map i the tableau o is the Fibonacci tableau. The restriction o such a polynomial to well chosen domains U 0 and U 1 gives rise to an L-map : U 0 U 1 V. It is then clear rom the deinition o the Fibonacci tableau that every critical piece o has exactly two children (this is in act the reason why the Fibonacci tableau was introduced by Branner and Hubbard), and that there are no central returns. Hence the L-map : U 0 U 1 V is a Fibonacci map in our sense. On the other hand, we will show that the orbit o the critical point o our Fibonacci maps returns closer to zero ater each Fibonacci number o iterations in some combinatorial sense; more precisely, we will show that S n (ω) belongs

14 14 X. BUFF V U 1 U 0 Figure 7. A Fibonacci map having a critical point o degree 6. to the the critical puzzle piece o depth S n+1 3 but not to the critical puzzle piece o depth S n+1 2. This is precisely the way Branner and Hubbard deine the Fibonacci tableau. This discussion and proposition 12.8 in [BH] show that there exist Fibonacci maps having a critical point o arbitrary degree l 2. In [SN], van Strien and Nowicki mimicked an argument due to Lyubich and Milnor [LM] and prove that or every even integer l 2, there exists a Fibonacci map : U 0 U 1 V satisying (z) = (z) and (z) = F (z l ) with F (0) 0. Deinition 14. We will say that is a real symmetric Fibonacci map o degree l i and only i : U 0 U 1 V is a Fibonacci map satisying (z) = (z) and (z) = F (z l ) with F (0) 0. The proo is based on the ormal machinery o kneading theory developed in [MT]. The irst step consists in constructing a polynomial P (z) = z l + c such that the orbit o the critical point returns closer to zero ater each Fibonacci number o iterations. The second step consists in renormalizing this polynomial in the sense o L-maps, so as to get an L-map : U 0 U 1 V where U 0 = P 5 and U 1 = P 3. One can easily check that or this map, the orbit o the critical point still returns closer to zero ater each Fibonacci number o iterations so that it is a Fibonacci map. We will now show that a Fibonacci map is ininitely renormalizable in the sense o L-maps. Aterwards, we will prove the equivalence between our deinition o Fibonacci maps and the one given by Branner and Hubbard Renormalization o Fibonacci maps.

15 FIXED POINTS OF RENORMALIZATION. 15 Proposition 6. (see picture 8) Given a Fibonacci map : U 0 U 1 V, let V 1 = C 0 be the critical piece o depth 0, U1 0 = C 2 be the critical piece o depth 2, and U1 1 = P 1 ( 2 (ω)) be the piece o depth 1 that contains 2 (ω). Then, the mapping g : U1 0 U1 1 V 1 deined by g U 0 1 = 2 and g U 1 1 = is a Fibonacci map. V U 0 U 1 (ω) 2 (ω) U 1 1 U 0 1 ω 2 Figure 8. A Fibonacci map is (2, 1) renormalizable. Remark. We will call g the canonical renormalization o. Proo. Let us irst prove a lemma that will be useul to prove this proposition and lemma 4 below. Lemma 1. Let : U 0 U 1 V be a Fibonacci map. Then, (ω) U 1 and i n (ω) U 1, we have (n+1) (ω) U 0. Proo. The irst statement simply ollows rom the act that has no central returns. The second statement ollows rom the act that C 1 = V has only two children. Indeed, let C 0 = U 0 and C 1 be the critical puzzle pieces o depth 0 and 1. Then, writing C 1 and C 1 U 1 C 1, C 0 we see that C 0 and C 1 are the two children o C 1. In particular, there can be no extra child. Thus, let U = 1 U 1 (U 1 ) be the connected component o 1 (U 1 ) which is contained in U 1 (see picture 8). Let us prove that the critical orbit never enters U. I this were not the case, then we could deine j to be the least integer such that j (ω) enters U, and we could pull-back univalently the puzzle piece U along the orbit (ω) j (ω) U. Pulling-back once more by U 0, we would obtain an extra child o C 1. Hence, i n (ω) U 1, then (n+1) (ω) V \ U 1 and since (n+1) (ω) K() U 0 U 1, we see that the proo o the lemma is completed.

16 16 X. BUFF Let us now prove that the map g : U1 0 U1 1 V 1 is an L-map. Since the connected components o the domain and range o g are puzzle pieces, they are topological disks, and given two o them, we have only three possible conigurations: they are equal, one is compactly contained in the other one, or their closures are disjoint. Since U1 0 is the critical piece o depth 2 and V 1 is the critical piece o depth 0, we see that U1 0 V 1. Furthermore, since (ω) U 1 and 2 (ω) U 0 = V 1, we see that 2 : U1 0 V 1 is a ramiied covering with, ramiied only at ω. On one hand, the no central returns condition implies that 2 (ω) U1 0. On the other hand, by deinition, 2 (ω) U1 1. Hence, the closures o U1 0 and U1 1 are disjoint. Besides, lemma 1 shows that V 1 contains 2 (ω). Since, V 1 is a puzzle piece o depth 0 and U1 1 is a puzzle piece o depth 1, and since both o them contain 2 (ω), we see that U1 1 V 1. Let us now show that the critical orbit o g never escapes U1 0 U1 1. Assume g n (ω) V 1 \ U1 0 = C 0 \ C 2. We want to show that g n (ω) U1 1. Since g n (ω) = k (ω) or some integer k, and since ω K(), we see that g n (ω) belongs to a puzzle piece o o depth 1 contained in C 0 \ C 2. It cannot be inside the critical piece C 1 since the puzzle pieces o depth 2 contained in C 1 \ C 2 are mapped by and 2 into U 1, so that (k+1) (ω) and (k+2) (ω) would both be inside U 1, contradicting lemma 1. Hence, g n (ω) is inside a puzzle piece o depth 1 contained in C 0 \ C 1. Assume it is a puzzle piece U U1 1. Then, we can use the same argument as in the proo o lemma 1: we let j be the least integer such that j (ω) enters U, and we pull-back C 0 along the ollowing orbits: ω (ω) 2 (ω) C 0, ω (ω) 2 (ω) U1 1 3 (ω) C 0 and ω (ω) j (ω) U (j+1) C 0 showing that C 0 has at least 3 children. Let us inally show that g : U1 0 U1 1 V 1 is a Fibonacci map. The critical pieces o the puzzle o g are exactly the critical pieces o the puzzle o which are children o C 0, grand-children o C 0, grand-grand-children o C 0 and so on. In particular, every critical piece o g has exactly two g-children. Besides, since has no central returns, the same property holds or g, which concludes the proo o the proposition. Let us now use this renormalization result to prove that our deinition o Fibonacci maps is equivalent to the one given by Branner and Hubbard. Proposition 7. I : U 0 U 1 V is a Fibonacci map, then or any n 0, Sn (ω) belongs to the critical puzzle piece o depth S n+1 3 but not to the critical piece o depth S n+1 2. Remark. One can easily check that this correspond to the deinition o the Fibonacci marked grid given in [BH], example Proo. Let us subdivide the proo within two lemmas that will be used again later. Lemma 2. Let : U 0 U 1 V be a Fibonacci map. Then, we can deine a sequence ( n : Un 0 Un 1 ) V n n 0 where 0 = and n+1 is the canonical renormalization o n. Then or n 1, the connected components o the range and the domain o n are V n = C Sn+1 3, U 0 n = C Sn+2 3 and U 1 n = P Sn+1+S n 1 3 ( S n (ω) ) ; the restrictions o n to U 0 n and U 1 n are n U 0 n = Sn and n U 1 n = Sn 1.

17 FIXED POINTS OF RENORMALIZATION. 17 Proo. The proo is an easy induction based on the deinition o the canonical renormalization. We leave the details to the reader. Lemma 3. Let : U 0 U 1 V be a Fibonacci map. Then, or any integer n 1, C Sn+1 2 is a child o C Sn 1 2. Proo. This is again proved by induction. We irst claim that the induction property holds or n = 1. Indeed, C S2 2 = C 1, C S0 2 = C 1 and we have C 1 U 1 C 1, so that C 1 is a child o C 1. Next, assume that the induction property holds or some integer n 1. Then, S n+1 restricts to a ramiied covering between the critical piece o depth S n+2 2 and the piece o depth S n+2 2 S n+1 = S n 2 which contains S n+1 (ω). Since lemma 2 shows that Sn+1 (ω) C Sn+2 3 C Sn 2, we see that Sn+1 : C Sn+2 2 C Sn 2 is a ramiied covering. We still need to see that S n+1 : C Sn+2 2 C Sn 2 is ramiied only at ω. To prove this result, observe that S n+1 = S n 1 S n. Since C Sn+2 2 C Sn+1 2 and since C Sn+1 2 is a child o C Sn 1 2, we see that Sn : C Sn+2 2 S n (C Sn+2 2) is a ramiied covering, ramiied only at ω. Hence, we only need to prove that the restriction o Sn 1 to Sn (C Sn+2 2) is univalent. We already know that this restriction is a (possibly ramiied) covering onto its image. Hence, we must show that S n (C Sn+2 2) does not contain a critical point o S n 1. Recall that by lemma 2, the restriction o Sn 1 to C Sn+1 3 has a unique critical point at ω. Hence, it is suicient to show that S n (C Sn+2 2) C Sn+1 3 \ C Sn+1 2 (see igure 9). By deinition, S n (C Sn+2 2) is the puzzle piece o depth S n+2 2 S n = S n C Sn 1 2 S n 1 S n 1 ω C Sn 2 C Sn+2 2 S n S n 1 (ω) Sn (ω) C Sn 3 C Sn+1 3 C Sn+1 2 Figure 9. The position o S n (C Sn+2 2).

18 18 X. BUFF S n+1 2 that contains S n (ω). By lemma 2, we have S n (ω) C Sn+1 3, so that Sn (C Sn+2 2) C Sn+1 3. Besides, the induction property at level n says that C Sn+1 2 is a child o C Sn 1 2 and the no central return condition implies that S n (ω) C Sn+1 2. In particular, S n (C Sn+2 2) C Sn+1 3 \ C Sn+1 2. The proo o the proposition is contained within the proo o lemma 3, since Sn (0) Sn (C Sn+2 2) C Sn+1 3 \ C Sn+1 2. From now on, all Fibonacci maps we will consider will be real symmetric Fibonacci maps. In particular, the critical point is 0. We now come back to renormalization o Fibonacci maps. Deinition 15. We can deine a renormalization operator R (2,1), on the set o Fibonacci maps, by R (2,1) () = α 1 g α, where g is the canonical renormalization o, and where α(z) = g(0) z = 2 (0) z. The map R (2,1) () is normalized so that its critical value is 1. This map can be projected to the space o germs o Fibonacci maps. We will keep the letter R (2,1) to denote the projection. The results obtained by van Strien and Nowicki in [SN] can be reormulated in the ollowing way. Theorem 3. (see [SN], Theorem 7.1) Assume 1 : U1 0 U1 1 V 1 and 2 : U2 0 U2 1 V 2 are two real symmetric Fibonacci maps o even degree l 4, such that 1 (0) 1 2 (0) and 2 (0) 2 2 (0) have the same sign. Then, there exists a quasi-conormal homeomorphism ψ : V 1 V 2 which conjugate the Fibonacci maps 1 and 2. Besides, there exists a constant ε > 0 such that ψ is C 1+ε at 0. For every even integer l 4, the renormalization operator R (2,1) has a unique cycle {[ 1 ], [ 2 ]} o order 2, where [ 1 ] and [ 2 ] are two germs o real symmetric Fibonacci maps o degree l. I : U 0 U 1 V is a real symmetric Fibonacci map o even degree l 4, then the sequence R n (2,1) ([]) converges to the cycle {[ 1 ], [ 2 ]}. Proo. The proo given by van Strien and Nowicky consists in irst obtaining real a-priori bounds which show that the closure o the post-critical set is a Cantor set with bounded geometry. I 1 (0) 1 2 (0) and 2 (0) 2 2 (0) have the same sign, then the ordering o the critical orbit on the real axis is the same, so that two maps are quasi-symmetrically conjugate along their critical orbit. Then, applying a pullback argument due to Sullivan and described in [dmvs], van Strien and Nowicky show that this quasi-symmetric conjugacy can be promoted to a quasi-conormal conjugacy between the two Fibonacci maps. To prove that the conjugacy is C 1+ε at 0, they use renormalization techniques, and the theory o towers introduced by McMullen in [McM2]. They show the convergence o renormalizations to a cycle o order 2 at the same time. Let us now improve this result in the ollowing way. Theorem A. For every even integer l 4, let i : U 0 i U 1 i V i, i = 1, 2, be the two real symmetric Fibonacci maps o degree l, normalized so that ω i = 0 and

19 FIXED POINTS OF RENORMALIZATION. 19 i (ω i ) = 1, and satisying R (2,1) ([ 1 ]) = [ 2 ] and R (2,1) ([ 2 ]) = [ 1 ]. Then, there exists a neighborhood U o 0 and a neighborhood U o 1 such that 1 U U 0 1 = 2 U U 0 2, and 1 U U1 1 = 2 U U2 1. Proo. To prove this theorem, let us deine a lipping operator which to a Fibonacci map associates the new map : U 0 U 1 V deined by U 0 = and U 1 =. Lemma 4. I : U 0 U 1 V is a real symmetric Fibonacci map, then : U 0 U 1 V is still a Fibonacci map. Proo. The act is still an L-map is not obvious. We must irst show that the orbit o ω = 0 stays in U 0 U 1, and then show that the genealogical properties are satisied. We deine ω n to be the n-th iterate o the critical point: ω n = n (ω). Let us show by induction that or any n 0, ω n = n (ω) = ±ω n i ω n U 0 and ω n = ω n i ω n U 1. Indeed, we have already mentioned in the previous proo that the dierence between two consecutive returns o the critical orbit in U 0 is at most 2. This implies that i ω n 1 U 1, then ω n U 0. Hence, assuming the induction property holds or n 1, we see that i ω n 1 U 0, then ω n = (±ω n 1 ) = ω n U 0 U 1, and i ω n 1 U 1, then ω n = (ω n 1 ) = ω n U 0. This shows that the induction property is true or n. Besides, the critical pieces o the puzzle o are exactly the pieces Cn and k (C n ) is either k (C n ), or k (C n ). Thus, the genealogy o and o are exactly the same. Hence, is a Fibonacci map. Lemma 5. I : U 0 U 1 V is a real symmetric Fibonacci map, then the lipping operator and the renormalization operator commute: R (2,1) ([]) = R (2,1) ([ ]). Proo. We have seen that the central branch o the canonical renormalization o is 1 0, where 0 = U 0 and 1 = U 1. The other branch is 1. Hence, R (2,1) ([]) has central branch 1/α 1 0 (αz) and outer branch 1/α 1 (αz), where α = 2 (0). On the other hand, the canonical renormalization o has central branch 1 0 and outer branch 1, and 2 (0) = 2 (0) = α. Hence, R (2,1) ([ ]) has central branch 1/α [ 1 0 ( αz) ] = 1/α 1 0 (αz) and outer branch 1/α [ 1 ( αz) ] = [ 1/α 1 (αz) ]. We now claim that i {[ 1 ], [ 2 ]} is the cycle o order 2 o real symmetric germs o Fibonacci maps o degree l which is invariant by R (2,1), then we necessarily have [ 2 ] = [ 1 ]. Indeed, observe that {[ 1 ], [ 2 ]} is a cycle o order 2 o real symmetric Fibonacci maps o degree l which is invariant by R (2,1). By uniqueness o such a cycle (see theorem 3), we have {[ 1 ], [ 2 ]} = {[ 1 ], [ 2 ]}, and since [ 1 ] [ 1 ], we have [ 2 ] = [ 1 ]. This shows that 1 and 2 coincide in a neighborhood o K( 1 ) U1 0 = K( 2 ) U2 0, and 1 and 2 coincide in a neighborhood o K( 1 ) U1 1 = K( 2 ) U2 0, which concludes the proo o theorem A.

20 20 X. BUFF We will now show that i : U 0 U 1 V is a real symmetric Fibonacci map such that R (2,1) ([]) = [ ], then the restriction 0 = U 0 o to U 0 satisies the ollowing system o equations, that we will call the Cvitanović-Fibonacci equation: (z) = 1 (α(αz)), 0 < α < 1, α2 (0) = 1 and (z) = F (z l ), with F (0) 0 and l 4 even. Indeed, let 1 = U 1. Then, writing down R (2,1) ([]) = [ ] gives: { 0 (z) = 1/α 2 0 (α 0 (αz)), and 1 (z) = 1/α 0 (αz), Indeed, we have seen that the central branch o R (2,1) ([]) is 1/α 1 0 (αz) and the outer branch is 1/α 1 (αz). As R (2,1) ([]) = [ ], we get and 0 (z) = 1 α 1 0 (αz) 1 (z) = 1 α 0(αz) or all z in the Julia set K(), which enables us to conclude, replacing 1 in the irst equation Solutions o the Cvitanović-Fibonacci equation. We will now study the geometry o the solutions o the Cvitanović-Fibonacci equation. In particular, we will study the domain o analyticity o such solutions. Deinition 16. Let and g be two holomorphic unctions deined on open connected domains o C: U and U g. We say g is an analytic extension o i g is equal to on some non-empty open set. Moreover, i all such analytic extension are restriction o a single map ˆ : Ŵ C, we will say that ˆ is the maximal analytic extension o. Theorem B. For every even integer l 4, let be the solution o the Cvitanović- Fibonacci equation in degree l, and set α (z) = (αz) and α 2(z) = (α 2 z). Then, there exist domains W α C and W α 2 C containing 0 such that α : W α α (W α ) and α 2 : W α 2 α 2(W α 2) are polynomial-like mappings o degree l. Besides, α : W α α (W α ) has an attracting cycle o order 2 and α 2 : W α 2 α 2(W α 2) has an attracting ixed point. In particular, the Julia set J( α ) is quasi-conormally homeomorphic to the Julia set J(z z l 1) and the Julia set J( α 2) is a quasi-circle. Finally, the domain o analyticity o is the quasi-disk Ŵ bounded by the quasicircle αj( α 2). Remark. The unctions α and α 2 are not conjugated to. As we will see, their dynamical behavior is really dierent. Proo. The map α is a solution o the universal equation we introduced in the preceding chapter or λ = α 2 and ν = 1/2: α (z) = 1 λ α( α (λ 2 z)).

21 FIXED POINTS OF RENORMALIZATION. 21 We will use the notations o the preceding chapter, indexing all the sets with the letter α. We have seen that there exists a bounded domain W α C such that α : W α C α 2 = C(] 1/α 2, 1/α 4 [) is a ramiied covering. Let us now show that the interpretation o the universal equation in terms o a linearization equation enables us to prove the ollowing lemma. Lemma 6. (see igure 1) The maps α : W α C α 2 and α 2 : W α 2 = W α /α C α 2 are DH-maps with attracting cycles. Proo. By deinition o W α, the mapping α is a ramiied covering map rom W α to C α 2 with only one critical point o degree l in 0. Hence, the map α 2 is a ramiied covering rom W α 2 to C α 2. Moreover, as the degree is even, W α R = [ x 0 /α, x 0 /α] ] 1/α, 1/α[, with α (x 0 ) = 0. The inclusion is given by the relative position o points obtained rom igure 4. Hence, α and α 2 are both polynomial-like maps. We will show they are DH-maps, i.e., that the critical point does not escape. To do this, it is enough to show that α has an attracting cycle o period 2, and α 2 has an attracting ixed point. Those cycles must attract the critical point. We can rewrite the Cvitanović-Fibonacci equation, using the unctions α and α 2 in two dierent ways: α (z) = 1 α 2 α( α 2(z)) α 2(z) = 1 α 2 α( α ( α 2 z)) The irst equation tells us that α linearizes α 2 in a neighborhood o x 0 : x 0 α2 x 0 α 0 α 2 z Hence α 2 has an attracting ixed point, x 0, o multiplier α 2. The second equation tells us α α is conjugated by z α 2 z to α 2. Hence, α α has an attracting ixed point: α 2 x 0 < 0. Since α ( α 2 x 0 ) > 0, α has a cycle o period 2: { α 2 x 0, α ( α 2 x 0 )}. We have just shown that α 2 is a DH-map with an attracting ixed point. It ollows immediately that the attracting basin Ŵα o α 2 is a quasi-disk. Moreover, as α is the linearizer o α 2, it has a maximal analytic extension ˆ α : Ŵ α C. To conclude the proo, just remind that (z) = α (z/α), and deine Ŵ = αŵα. Remark. The basin o attraction o the DH-map α 2 : W α 2 C α 2 being Ŵ /α, and α 2 being conjugate to α 2 by the scaling map z α 2 z, we see that the immediate basin o the DH-map α : W α C α 2 has two connected components, the one containing 0 being αŵ, the other one being α(αŵ ). Beore going urther, let us observe some consequences o this theorem. The ollowing lemma will be useul in the construction o a particular Fibonacci map (see theorem C). 0. α

22 22 X. BUFF Lemma 7. The mapping α : Ŵ Ŵα is a DH-map representing the same germ as α : Ŵ α C α 2. Proo. Since Ŵα is by deinition the basin o attraction o the DH-map α 2 : W α 2 C α 2, we see that the map α 2 : Ŵ α Ŵα is a ramiied covering, ramiied only at 0. Hence, the same property holds or the map α : Ŵ Ŵα. Let us now show that Ŵ is relatively compact in Ŵα. This is an immediate consequence o the ollowing inclusion o sets (see igure 10): W Ŵ W α = W/α. The irst inclusion is obvious because is analytic on W, which has to be inside Ŵ W α = W/α W Figure 10. We have W Ŵ W α = W/α. the domain o analyticity Ŵ o. To show the second inclusion, note that the map α 2 : W α 2 C α 2 is a DH-map. Hence its illed-in Julia set, i.e., the closure o Ŵα, is contained in W α 2. This concludes the proo o the lemma, since any polynomial-like restriction o a DH-map represent the same germ. Deinition 17. For k 1, we deine D k and D k to be the sets D k = { z Ŵα (k+1) α 2 (z) Ŵ } and D k = { z Ŵ (k+1) α (z) Ŵ }. We can now prove the ollowing geometric result, which will be used in the proos o theorems C and D.

23 FIXED POINTS OF RENORMALIZATION. 23 Lemma 8. For any k 1, the sets D k and D k are quasi-disks. We have the inclusions D k D k+1 Ŵα and K( α ) D k+1 D k. Besides, the mappings α 2 : D k+1 D k and α : D k+1 D k are ramiied coverings, ramiied only at 0. Furthermore, the closure o the set D k resp. D k converges, or the Hausdor topology on compact subsets o P 1, to the illed-in Julia set K( α 2) resp. K( α ) as k tends to ininity. Proo. The statement or the sets D k is an immediate consequence o the act that α : Ŵ Ŵα is a DH-map whose Julia set is K( α ), combined with the act that D k = [ α Ŵ ] (k+1) (Ŵ ). Proving the statement or the sets D k is o the same order o diiculty. Indeed, the set Ŵ = D 1 is contained in the basin o attraction o α 2, i.e. Ŵα. Besides, D k = [ α 2 Ŵα ] (k+1) (Ŵ ). Finally, α 2(Ŵ ) = α(αŵ ), and since the immediate basin o the DH-map α : Ŵ Ŵ α is αŵ α(αŵ ), we see that α 2(Ŵ ) Ŵ Construction o a Fibonacci map. We will now prove that any solution o the Cvitanović-Fibonacci equation gives rise to a cycle o order 2 o Fibonacci maps which is invariant under renormalization. Theorem C. (see igure 11) Given any solution o the Cvitanović-Fibonacci equation, there exists a Fibonacci map φ : U 0 U 1 V such that φ and coincide on U 0 and such that R 2 (2,1)([φ]) = [φ]. Proo. We irst need to deine the map φ. We deine V to be the domain Ŵ. The domain U 0 is deined to be equal to α 2 D 0 (deined in deinition 17). Combining lemmas 1 and 7, we see that the mapping α : Ŵ Ŵα is a DH-map having a cycle o period 2: { α 2 x 0, α ( α 2 x 0 )}. The immediate basin o this cycle has two connected components. The one containing 0 is αŵ. We deine U 1 to be the other connected component. It is clear that U 0, U 1 and V are quasi-disks. We claim that the map φ : U 0 U 1 V, deined by { φ U 0(z) = (z) φ U 1(z) = 1 α (αz), is a Fibonacci map, and that the germ [φ] is a ixed point o R (2,1). Step 1. Let us irst show that U 0 and U 1 are disjoint and contained in V. Lemma 8 says that D 0 = U 0 /α 2 is compactly contained in Ŵα = Ŵ /α. Hence, the closure o U 0 is contained in αŵ. Since the immediate basin o the DH-map α : Ŵ Ŵα is αŵ U 1, we see that U 0 and U 1 are disjoint and contained in V = Ŵ (see igure 12). Step 2. The map φ : U 0 V is a ramiied covering, ramiied only at 0 and the map φ : U 1 V is an isomorphism. Indeed, lemma 8 states that α 2 : D 0 D 1 is a ramiied covering ramiied only at 0. Using α 2(z) = (α 2 z), D 0 = U 0 /α 2 and D 1 = V, the irst statement is proved. Using again that the immediate basin o the DH-map α : Ŵ Ŵα is αŵ U 1, and that the critical point o this

24 24 X. BUFF φ U 0(z) = (z) V φ U 1(z) = 1 α (αz) U 0 U 1 Figure 11. The Fibonacci map φ. DH-map is contained in αŵ, we see that α : U 1 αŵ is an isomorphism, which immediately implies the second statement, since φ U 1 = α /α. Step 3. We now need to show that the critical orbit does not escape rom U 0 U 1. For this purpose, we will prove a result that will be used again later, in the study o the shape o puzzle pieces. In the ollowing lemma, [x] denotes the integer part o x. Lemma 9. For any k 0, φ S 2k is well deined on α 2k+2 D k and φ S 2k+1 is well deined on α 2k+2 Ŵ. Besides, φ S 2k : α 2k+2 D k α 2k D k 1 and φ S 2k+1 : α 2k+2 Ŵ α 2k Ŵ are ramiied coverings ramiied only at 0. In both cases, the iterate φ S n coincides with the map ( z ) z ( 1) [(n+1)/2] α n α n. Proo. We irst claim that this property holds or k = 0. Indeed, it says that φ is well deined on α 2 D 0 = U 0, φ : α 2 D 0 = U 0 D 1 = V is a ramiied covering ramiied only at 0 which coincides with ; φ 2 is well deined on α 2 Ŵ, and φ 2 : α 2 Ŵ Ŵ is a ramiied covering ramiied only at 0 which coincides with α(z/α). The irst point is obvious (by deinition o φ). The second point requires some argumentation. To prove this, remember that the immediate basin o the DH-map

Geometry of the Feigenbaum map. by Xavier Buff Université Paul Sabatier, Laboratoire Emile Picard, Toulouse cedex France. Abstract.

Geometry of the Feigenbaum map. by Xavier Buff Université Paul Sabatier, Laboratoire Emile Picard, Toulouse cedex France. Abstract. Geometry of the Feigenbaum map. by Xavier Buff Université Paul Sabatier, Laboratoire Emile Picard, 31062 Toulouse cedex France Abstract. We show that the Cvitanović-Feigenbaum equation can be interpreted

More information

SEPARATED AND PROPER MORPHISMS

SEPARATED AND PROPER MORPHISMS SEPARATED AND PROPER MORPHISMS BRIAN OSSERMAN The notions o separatedness and properness are the algebraic geometry analogues o the Hausdor condition and compactness in topology. For varieties over the

More information

SEPARATED AND PROPER MORPHISMS

SEPARATED AND PROPER MORPHISMS SEPARATED AND PROPER MORPHISMS BRIAN OSSERMAN Last quarter, we introduced the closed diagonal condition or a prevariety to be a prevariety, and the universally closed condition or a variety to be complete.

More information

VALUATIVE CRITERIA BRIAN OSSERMAN

VALUATIVE CRITERIA BRIAN OSSERMAN VALUATIVE CRITERIA BRIAN OSSERMAN Intuitively, one can think o separatedness as (a relative version o) uniqueness o limits, and properness as (a relative version o) existence o (unique) limits. It is not

More information

VALUATIVE CRITERIA FOR SEPARATED AND PROPER MORPHISMS

VALUATIVE CRITERIA FOR SEPARATED AND PROPER MORPHISMS VALUATIVE CRITERIA FOR SEPARATED AND PROPER MORPHISMS BRIAN OSSERMAN Recall that or prevarieties, we had criteria or being a variety or or being complete in terms o existence and uniqueness o limits, where

More information

CHOW S LEMMA. Matthew Emerton

CHOW S LEMMA. Matthew Emerton CHOW LEMMA Matthew Emerton The aim o this note is to prove the ollowing orm o Chow s Lemma: uppose that : is a separated inite type morphism o Noetherian schemes. Then (or some suiciently large n) there

More information

Return times of polynomials as meta-fibonacci numbers

Return times of polynomials as meta-fibonacci numbers Return times of polynomials as meta-fibonacci numbers arxiv:math/0411180v2 [mathds] 2 Nov 2007 Nathaniel D Emerson February 17, 2009 University of Southern California Los Angeles, California 90089 E-mail:

More information

arxiv:math/ v1 [math.ds] 15 Jun 1996

arxiv:math/ v1 [math.ds] 15 Jun 1996 Parameter Scaling for the Fibonacci Point arxiv:math/9606218v1 [math.ds] 15 Jun 1996 LeRoy Wenstrom Mathematics Department S.U.N.Y. Stony Brook Abstract We prove geometric and scaling results for the real

More information

Hyperbolicity of Renormalization for C r Unimodal Maps

Hyperbolicity of Renormalization for C r Unimodal Maps Hyperbolicity of Renormalization for C r Unimodal Maps A guided tour Edson de Faria Department of Mathematics IME-USP December 13, 2010 (joint work with W. de Melo and A. Pinto) Ann. of Math. 164(2006),

More information

Rigidity for real polynomials

Rigidity for real polynomials Rigidity for real polynomials O. Kozlovski, W. Shen, S. van Strien June 6, 2003 Abstract We prove the topological (or combinatorial) rigidity property for real polynomials with all critical points real

More information

( x) f = where P and Q are polynomials.

( x) f = where P and Q are polynomials. 9.8 Graphing Rational Functions Lets begin with a deinition. Deinition: Rational Function A rational unction is a unction o the orm ( ) ( ) ( ) P where P and Q are polynomials. Q An eample o a simple rational

More information

Math 216A. A gluing construction of Proj(S)

Math 216A. A gluing construction of Proj(S) Math 216A. A gluing construction o Proj(S) 1. Some basic deinitions Let S = n 0 S n be an N-graded ring (we ollows French terminology here, even though outside o France it is commonly accepted that N does

More information

(C) The rationals and the reals as linearly ordered sets. Contents. 1 The characterizing results

(C) The rationals and the reals as linearly ordered sets. Contents. 1 The characterizing results (C) The rationals and the reals as linearly ordered sets We know that both Q and R are something special. When we think about about either o these we usually view it as a ield, or at least some kind o

More information

2 Genadi Levin and Sebastian van Strien of the Mandelbrot set, see [Y] (Actually, McMullen proved that whenever the critical orbit of an innitely reno

2 Genadi Levin and Sebastian van Strien of the Mandelbrot set, see [Y] (Actually, McMullen proved that whenever the critical orbit of an innitely reno Local connectivity of the Julia set of real polynomials Genadi Levin, Hebrew University, Israel Sebastian van Strien, University of Amsterdam, the Netherlands y December 31, 1994 and extended January 27,

More information

QUADRATIC JULIA SETS WITH POSITIVE AREA.

QUADRATIC JULIA SETS WITH POSITIVE AREA. QUADRATIC JULIA SETS WITH POSITIVE AREA. XAVIER BUFF AND ARNAUD CHÉRITAT Abstract. We prove the existence o quadratic polynomials having a Julia set with positive Lebesgue measure. We ind such examples

More information

In this paper we consider complex analytic rational maps of the form. F λ (z) = z 2 + c + λ z 2

In this paper we consider complex analytic rational maps of the form. F λ (z) = z 2 + c + λ z 2 Rabbits, Basilicas, and Other Julia Sets Wrapped in Sierpinski Carpets Paul Blanchard, Robert L. Devaney, Antonio Garijo, Sebastian M. Marotta, Elizabeth D. Russell 1 Introduction In this paper we consider

More information

Problem Set. Problems on Unordered Summation. Math 5323, Fall Februray 15, 2001 ANSWERS

Problem Set. Problems on Unordered Summation. Math 5323, Fall Februray 15, 2001 ANSWERS Problem Set Problems on Unordered Summation Math 5323, Fall 2001 Februray 15, 2001 ANSWERS i 1 Unordered Sums o Real Terms In calculus and real analysis, one deines the convergence o an ininite series

More information

Bifurcation of Unimodal Maps *

Bifurcation of Unimodal Maps * QUALITATIVE THEORY OF DYNAMICAL SYSTEMS 4, 413 424 (2004) ARTICLE NO. 69 Bifurcation of Unimodal Maps * Welington de Melo Instituto Nacional de Matemática Pura e Aplicada, Estrada Dona Castorina 110, 22460-320,

More information

Dynamics on Hubbard trees

Dynamics on Hubbard trees FUNDAME NTA MATHEMATICAE 164(2000) Dynamics on Hubbard trees by Lluís Alsedà and Núria Fagella (Barcelona) Abstract. It is well known that the Hubbard tree of a postcritically finite complex polynomial

More information

AN INEQUALITY FOR LAMINATIONS, JULIA SETS AND GROWING TREES. A. Blokh and G. Levin

AN INEQUALITY FOR LAMINATIONS, JULIA SETS AND GROWING TREES. A. Blokh and G. Levin AN INEQUALITY FOR LAMINATIONS, JULIA SETS AND GROWING TREES A. Blokh and G. Levin Abstract. For a closed lamination on the unit circle invariant under z z d we prove an inequality relating the number of

More information

Math 754 Chapter III: Fiber bundles. Classifying spaces. Applications

Math 754 Chapter III: Fiber bundles. Classifying spaces. Applications Math 754 Chapter III: Fiber bundles. Classiying spaces. Applications Laurențiu Maxim Department o Mathematics University o Wisconsin maxim@math.wisc.edu April 18, 2018 Contents 1 Fiber bundles 2 2 Principle

More information

On the regular leaf space of the cauliflower

On the regular leaf space of the cauliflower On the regular leaf space of the cauliflower Tomoki Kawahira Department of Mathematics Graduate School of Science Kyoto University Email: kawahira@math.kyoto-u.ac.jp June 4, 2003 Abstract We construct

More information

ON KÖNIG S ROOT-FINDING ALGORITHMS.

ON KÖNIG S ROOT-FINDING ALGORITHMS. ON KÖNIG S ROOT-FINDING ALGORITHMS. XAVIER BUFF AND CHRISTIAN HENRIKSEN Abstract. In this article, we first recall the definition of a family of rootfinding algorithms known as König s algorithms. We establish

More information

Non Locally-Connected Julia Sets constructed by iterated tuning

Non Locally-Connected Julia Sets constructed by iterated tuning Non Locally-Connected Julia Sets constructed by iterated tuning John Milnor Stony Brook University Revised May 26, 2006 Notations: Every quadratic map f c (z) = z 2 + c has two fixed points, α and β, where

More information

The Dynamics of Two and Three Circle Inversion Daniel M. Look Indiana University of Pennsylvania

The Dynamics of Two and Three Circle Inversion Daniel M. Look Indiana University of Pennsylvania The Dynamics of Two and Three Circle Inversion Daniel M. Look Indiana University of Pennsylvania AMS Subject Classification: Primary: 37F10 Secondary: 51N05, 54D70 Key Words: Julia Set, Complex Dynamics,

More information

ADMISSIBILITY OF KNEADING SEQUENCES AND STRUCTURE OF HUBBARD TREES FOR QUADRATIC POLYNOMIALS

ADMISSIBILITY OF KNEADING SEQUENCES AND STRUCTURE OF HUBBARD TREES FOR QUADRATIC POLYNOMIALS ADMISSIBILITY OF KNEADING SEQUENCES AND STRUCTURE OF HUBBARD TREES FOR QUADRATIC POLYNOMIALS HENK BRUIN AND DIERK SCHLEICHER Abstract. Hubbard trees are invariant trees connecting the points of the critical

More information

TOEPLITZ KNEADING SEQUENCES AND ADDING MACHINES

TOEPLITZ KNEADING SEQUENCES AND ADDING MACHINES TOEPLITZ KNEADING SEQUENCES AND ADDING MACHINES LORI ALVIN Department of Mathematics and Statistics University of West Florida 11000 University Parkway Pensacola, FL 32514, USA Abstract. In this paper

More information

Roberto s Notes on Differential Calculus Chapter 8: Graphical analysis Section 1. Extreme points

Roberto s Notes on Differential Calculus Chapter 8: Graphical analysis Section 1. Extreme points Roberto s Notes on Dierential Calculus Chapter 8: Graphical analysis Section 1 Extreme points What you need to know already: How to solve basic algebraic and trigonometric equations. All basic techniques

More information

Puiseux series dynamics and leading monomials of escape regions

Puiseux series dynamics and leading monomials of escape regions Puiseux series dynamics and leading monomials of escape regions Jan Kiwi PUC, Chile Workshop on Moduli Spaces Associated to Dynamical Systems ICERM Providence, April 17, 2012 Parameter space Monic centered

More information

Hyperbolic Component Boundaries

Hyperbolic Component Boundaries Hyperbolic Component Boundaries John Milnor Stony Brook University Gyeongju, August 23, 2014 Revised version. The conjectures on page 16 were problematic, and have been corrected. The Problem Hyperbolic

More information

Supplementary material for Continuous-action planning for discounted infinite-horizon nonlinear optimal control with Lipschitz values

Supplementary material for Continuous-action planning for discounted infinite-horizon nonlinear optimal control with Lipschitz values Supplementary material or Continuous-action planning or discounted ininite-horizon nonlinear optimal control with Lipschitz values List o main notations x, X, u, U state, state space, action, action space,

More information

This is a preprint of: On a Family of Rational Perturbations of the Doubling Map, Jordi Canela, Nuria Fagella, Antoni Garijo, J. Difference Equ.

This is a preprint of: On a Family of Rational Perturbations of the Doubling Map, Jordi Canela, Nuria Fagella, Antoni Garijo, J. Difference Equ. This is a preprint of: On a Family of Rational Perturbations of the Doubling Map, Jordi Canela, Nuria Fagella, Antoni Garijo, J. Difference Equ. Appl., vol. 21(8), 715 741, 2015. DOI: [10.1080/10236198.2015.1050387]

More information

UMS 7/2/14. Nawaz John Sultani. July 12, Abstract

UMS 7/2/14. Nawaz John Sultani. July 12, Abstract UMS 7/2/14 Nawaz John Sultani July 12, 2014 Notes or July, 2 2014 UMS lecture Abstract 1 Quick Review o Universals Deinition 1.1. I S : D C is a unctor and c an object o C, a universal arrow rom c to S

More information

Quadratic Polynomials and Combinatorics of the Principal Nest

Quadratic Polynomials and Combinatorics of the Principal Nest Quadratic Polynomials and Combinatorics of the Principal Nest RODRIGO A. PÉREZ To my son, Victor Alexander Perez ABSTRACT. We supplement the definition of principal nest, introduced by M. Lyubich, with

More information

EXTERNAL RAYS AND THE REAL SLICE OF THE MANDELBROT SET

EXTERNAL RAYS AND THE REAL SLICE OF THE MANDELBROT SET EXTERNAL RAYS AND THE REAL SLICE OF THE MANDELBROT SET SAEED ZAKERI Abstract. This paper investigates the set of angles of the parameter rays which land on the real slice [ 2, 1/4] of the Mandelbrot set.

More information

Local connectivity of Julia sets for unicritical polynomials

Local connectivity of Julia sets for unicritical polynomials ANNALS OF MATHEMATICS Local connectivity of Julia sets for unicritical polynomials By Jeremy Kahn and Mikhail Lyubich SECOND SERIES, VOL. 170, NO. 1 July, 2009 anmaah Annals of Mathematics, 170 (2009),

More information

3. The Sheaf of Regular Functions

3. The Sheaf of Regular Functions 24 Andreas Gathmann 3. The Sheaf of Regular Functions After having defined affine varieties, our next goal must be to say what kind of maps between them we want to consider as morphisms, i. e. as nice

More information

arxiv: v1 [math.ds] 13 Oct 2017

arxiv: v1 [math.ds] 13 Oct 2017 INVISIBLE TRICORNS IN REAL SLICES OF RATIONAL MAPS arxiv:1710.05071v1 [math.ds] 13 Oct 2017 RUSSELL LODGE AND SABYASACHI MUKHERJEE Abstract. One of the conspicuous features of real slices of bicritical

More information

Rational Maps with Generalized Sierpinski Gasket Julia Sets

Rational Maps with Generalized Sierpinski Gasket Julia Sets Rational Maps with Generalized Sierpinski Gasket Julia Sets Robert L. Devaney Department of Mathematics Boston University Boston, MA 02215, USA Mónica Moreno Rocha The Fields Institute University of Toronto

More information

QUADRATIC SIEGEL DISKS WITH SMOOTH BOUNDARIES.

QUADRATIC SIEGEL DISKS WITH SMOOTH BOUNDARIES. QUADRATIC SIEGEL DISKS WITH SMOOTH BOUNDARIES. XAVIER BUFF AND ARNAUD CHÉRITAT Abstract. We show the existence of angles α R/Z such that the quadratic polynomial P α(z) = e 2iπα z+z 2 has a Siegel disk

More information

arxiv: v2 [math.ds] 13 Sep 2017

arxiv: v2 [math.ds] 13 Sep 2017 DYNAMICS ON TREES OF SPHERES MATTHIEU ARFEUX arxiv:1406.6347v2 [math.ds] 13 Sep 2017 Abstract. We introduce the notion of dynamically marked rational map and study sequences of analytic conjugacy classes

More information

Theorem Let J and f be as in the previous theorem. Then for any w 0 Int(J), f(z) (z w 0 ) n+1

Theorem Let J and f be as in the previous theorem. Then for any w 0 Int(J), f(z) (z w 0 ) n+1 (w) Second, since lim z w z w z w δ. Thus, i r δ, then z w =r (w) z w = (w), there exist δ, M > 0 such that (w) z w M i dz ML({ z w = r}) = M2πr, which tends to 0 as r 0. This shows that g = 2πi(w), which

More information

Wandering Domains in Spherically Complete Fields

Wandering Domains in Spherically Complete Fields Wandering Domains in Spherically Complete Fields Eugenio Trucco Universidad Austral de Chile p-adic and Complex Dynamics, ICERM 20120214 Complex Polynomial Dynamics To understand the dynamics of a polynomial

More information

1 Relative degree and local normal forms

1 Relative degree and local normal forms THE ZERO DYNAMICS OF A NONLINEAR SYSTEM 1 Relative degree and local normal orms The purpose o this Section is to show how single-input single-output nonlinear systems can be locally given, by means o a

More information

If one wants to study iterations of functions or mappings,

If one wants to study iterations of functions or mappings, The Mandelbrot Set And Its Julia Sets If one wants to study iterations of functions or mappings, f n = f f, as n becomes arbitrarily large then Julia sets are an important tool. They show up as the boundaries

More information

Near-parabolic Renormalization and Rigidity

Near-parabolic Renormalization and Rigidity Near-parabolic enormalization and igidity Mitsuhiro Shishikura Kyoto University Complex Dynamics and elated Topics esearch Institute for Mathematical Sciences, Kyoto University September 3, 2007 Irrationally

More information

Evolution of the McMullen Domain for Singularly Perturbed Rational Maps

Evolution of the McMullen Domain for Singularly Perturbed Rational Maps Volume 32, 2008 Pages 301 320 http://topology.auburn.edu/tp/ Evolution of the McMullen Domain for Singularly Perturbed Rational Maps by Robert L. Devaney and Sebastian M. Marotta Electronically published

More information

Categories and Natural Transformations

Categories and Natural Transformations Categories and Natural Transormations Ethan Jerzak 17 August 2007 1 Introduction The motivation or studying Category Theory is to ormalise the underlying similarities between a broad range o mathematical

More information

STAT 801: Mathematical Statistics. Hypothesis Testing

STAT 801: Mathematical Statistics. Hypothesis Testing STAT 801: Mathematical Statistics Hypothesis Testing Hypothesis testing: a statistical problem where you must choose, on the basis o data X, between two alternatives. We ormalize this as the problem o

More information

MATH 8253 ALGEBRAIC GEOMETRY WEEK 12

MATH 8253 ALGEBRAIC GEOMETRY WEEK 12 MATH 8253 ALGEBRAIC GEOMETRY WEEK 2 CİHAN BAHRAN 3.2.. Let Y be a Noetherian scheme. Show that any Y -scheme X of finite type is Noetherian. Moreover, if Y is of finite dimension, then so is X. Write f

More information

THE GEOMETRY OF THE CRITICALLY-PERIODIC CURVES IN THE SPACE OF CUBIC POLYNOMIALS

THE GEOMETRY OF THE CRITICALLY-PERIODIC CURVES IN THE SPACE OF CUBIC POLYNOMIALS THE GEOMETRY OF THE CRITICALLY-PERIODIC CURVES IN THE SPACE OF CUBIC POLYNOMIALS LAURA DE MARCO AND AARON SCHIFF Abstract. We provide an algorithm for computing the Euler characteristic of the curves S

More information

Limiting Dynamics of quadratic polynomials and Parabolic Blowups

Limiting Dynamics of quadratic polynomials and Parabolic Blowups Limiting Dynamics of quadratic polynomials and Parabolic Blowups John Hubbard Joint work with Ismael Bachy MUCH inspired by work of Adam Epstein and helped by discussions with Xavier Buff and Sarah Koch

More information

2. ETA EVALUATIONS USING WEBER FUNCTIONS. Introduction

2. ETA EVALUATIONS USING WEBER FUNCTIONS. Introduction . ETA EVALUATIONS USING WEBER FUNCTIONS Introduction So ar we have seen some o the methods or providing eta evaluations that appear in the literature and we have seen some o the interesting properties

More information

Scaling ratios and triangles in Siegel disks. by

Scaling ratios and triangles in Siegel disks. by Scaling ratios and triangles in Siegel disks. by Xavier Buff Christian Henriksen. Université Paul Sabatier The Technical University of Denmark Laboratoire Emile Picard and Department of Mathematics 31062

More information

CS 361 Meeting 28 11/14/18

CS 361 Meeting 28 11/14/18 CS 361 Meeting 28 11/14/18 Announcements 1. Homework 9 due Friday Computation Histories 1. Some very interesting proos o undecidability rely on the technique o constructing a language that describes the

More information

Teichmüller distance for some polynomial-like maps

Teichmüller distance for some polynomial-like maps Teichmüller distance for some polynomial-like maps arxiv:math/9601214v1 [math.ds] 15 Jan 1996 1 Introduction Eduardo A. Prado Instituto de Matemática e Estatística Universidade de São Paulo Caixa Postal

More information

QUASISYMMETRIC RIGIDITY IN ONE-DIMENSIONAL DYNAMICS May 11, 2018

QUASISYMMETRIC RIGIDITY IN ONE-DIMENSIONAL DYNAMICS May 11, 2018 QUASISYMMETRIC RIGIDITY IN ONE-DIMENSIONAL DYNAMICS May 11, 2018 TREVOR CLARK AND SEBASTIAN VAN STRIEN Abstract In the late 1980 s Sullivan initiated a programme to prove quasisymmetric rigidity in one-dimensional

More information

COMBINATORIAL RIGIDITY FOR UNICRITICAL POLYNOMIALS

COMBINATORIAL RIGIDITY FOR UNICRITICAL POLYNOMIALS COMBINATORIAL RIGIDITY FOR UNICRITICAL POLYNOMIALS ARTUR AVILA, JEREMY KAHN, MIKHAIL LYUBICH AND WEIXIAO SHEN Abstract. We prove that any unicritical polynomial f c : z z d + c which is at most finitely

More information

Math 248B. Base change morphisms

Math 248B. Base change morphisms Math 248B. Base change morphisms 1. Motivation A basic operation with shea cohomology is pullback. For a continuous map o topological spaces : X X and an abelian shea F on X with (topological) pullback

More information

INVISIBLE TRICORNS IN REAL SLICES OF RATIONAL MAPS

INVISIBLE TRICORNS IN REAL SLICES OF RATIONAL MAPS INVISIBLE TRICORNS IN REAL SLICES OF RATIONAL MAPS RUSSELL LODGE AND SABYASACHI MUKHERJEE Abstract. One of the conspicuous features of real slices of bicritical rational maps is the existence of tricorn-type

More information

THE GEOMETRY OF THE CRITICALLY-PERIODIC CURVES IN THE SPACE OF CUBIC POLYNOMIALS

THE GEOMETRY OF THE CRITICALLY-PERIODIC CURVES IN THE SPACE OF CUBIC POLYNOMIALS THE GEOMETRY OF THE CRITICALLY-PERIODIC CURVES IN THE SPACE OF CUBIC POLYNOMIALS LAURA DE MARCO AND AARON SCHIFF Abstract. We provide an algorithm for computing the Euler characteristic of the curves S

More information

Teichmuller space of Fibonacci maps. March 18, 1993 Mikhail Lyubich Mathematics Department and IMS, SUNY Stony Brook

Teichmuller space of Fibonacci maps. March 18, 1993 Mikhail Lyubich Mathematics Department and IMS, SUNY Stony Brook Teichmuller space of Fibonacci maps. March 8, 993 Mikhail Lyubich Mathematics Department and IMS, SUNY Stony Brook x. Introduction. According to Sullivan, a space E of unimodal maps with the same combinatorics

More information

Topology of the regular part for infinitely renormalizable quadratic polynomials

Topology of the regular part for infinitely renormalizable quadratic polynomials Topology of the regular part for infinitely renormalizable quadratic polynomials Carlos Cabrera and Tomoki Kawahira March 16, 2010 Abstract In this paper we describe the well studied process of renormalization

More information

LECTURE 2. defined recursively by x i+1 := f λ (x i ) with starting point x 0 = 1/2. If we plot the set of accumulation points of P λ, that is,

LECTURE 2. defined recursively by x i+1 := f λ (x i ) with starting point x 0 = 1/2. If we plot the set of accumulation points of P λ, that is, LECTURE 2 1. Rational maps Last time, we considered the dynamical system obtained by iterating the map x f λ λx(1 x). We were mainly interested in cases where the orbit of the critical point was periodic.

More information

Julia Sets and the Mandelbrot Set

Julia Sets and the Mandelbrot Set Julia Sets and the Mandelbrot Set Julia sets are certain fractal sets in the complex plane that arise from the dynamics of complex polynomials. These notes give a brief introduction to Julia sets and explore

More information

Part V. 17 Introduction: What are measures and why measurable sets. Lebesgue Integration Theory

Part V. 17 Introduction: What are measures and why measurable sets. Lebesgue Integration Theory Part V 7 Introduction: What are measures and why measurable sets Lebesgue Integration Theory Definition 7. (Preliminary). A measure on a set is a function :2 [ ] such that. () = 2. If { } = is a finite

More information

An alternative proof of Mañé s theorem on non-expanding Julia sets

An alternative proof of Mañé s theorem on non-expanding Julia sets An alternative proof of Mañé s theorem on non-expanding Julia sets Mitsuhiro Shishikura and Tan Lei Abstract We give a proof of the following theorem of Mañé: A forward invariant compact set in the Julia

More information

Combinatorial rigidity of multicritical maps

Combinatorial rigidity of multicritical maps Combinatorial rigidity of multicritical maps Wenjuan Peng, Lei an o cite this version: Wenjuan Peng, Lei an. Combinatorial rigidity of multicritical maps. 30 pages. 2011. HAL Id: hal-00601631

More information

Singular Perturbations in the McMullen Domain

Singular Perturbations in the McMullen Domain Singular Perturbations in the McMullen Domain Robert L. Devaney Sebastian M. Marotta Department of Mathematics Boston University January 5, 2008 Abstract In this paper we study the dynamics of the family

More information

Descent on the étale site Wouter Zomervrucht, October 14, 2014

Descent on the étale site Wouter Zomervrucht, October 14, 2014 Descent on the étale site Wouter Zomervrucht, October 14, 2014 We treat two eatures o the étale site: descent o morphisms and descent o quasi-coherent sheaves. All will also be true on the larger pp and

More information

LECTURE 6: FIBER BUNDLES

LECTURE 6: FIBER BUNDLES LECTURE 6: FIBER BUNDLES In this section we will introduce the interesting class o ibrations given by iber bundles. Fiber bundles lay an imortant role in many geometric contexts. For examle, the Grassmaniann

More information

Math 145. Codimension

Math 145. Codimension Math 145. Codimension 1. Main result and some interesting examples In class we have seen that the dimension theory of an affine variety (irreducible!) is linked to the structure of the function field in

More information

arxiv: v1 [math.gt] 20 Feb 2008

arxiv: v1 [math.gt] 20 Feb 2008 EQUIVALENCE OF REAL MILNOR S FIBRATIONS FOR QUASI HOMOGENEOUS SINGULARITIES arxiv:0802.2746v1 [math.gt] 20 Feb 2008 ARAÚJO DOS SANTOS, R. Abstract. We are going to use the Euler s vector ields in order

More information

APPROXIMABILITY OF DYNAMICAL SYSTEMS BETWEEN TREES OF SPHERES

APPROXIMABILITY OF DYNAMICAL SYSTEMS BETWEEN TREES OF SPHERES APPROXIMABILITY OF DYNAMICAL SYSTEMS BETWEEN TREES OF SPHERES MATTHIEU ARFEUX arxiv:1408.2118v2 [math.ds] 13 Sep 2017 Abstract. We study sequences of analytic conjugacy classes of rational maps which diverge

More information

The Mandelbrot Set. Andrew Brown. April 14, 2008

The Mandelbrot Set. Andrew Brown. April 14, 2008 The Mandelbrot Set Andrew Brown April 14, 2008 The Mandelbrot Set and other Fractals are Cool But What are They? To understand Fractals, we must first understand some things about iterated polynomials

More information

HAUSDORFF DIMENSION OF BIACCESSIBLE ANGLES FOR QUADRATIC POLYNOMIALS. VERSION OF May 8, Introduction

HAUSDORFF DIMENSION OF BIACCESSIBLE ANGLES FOR QUADRATIC POLYNOMIALS. VERSION OF May 8, Introduction HAUSDORFF DIMENSION OF BIACCESSIBLE ANGLES FOR QUADRATIC POLYNOMIALS. VERSION OF May 8, 2017 HENK BRUIN AND DIERK SCHLEICHER Abstract. A point c in the Mandelbrot set is called biaccessible if two parameter

More information

Analytic continuation in several complex variables

Analytic continuation in several complex variables Analytic continuation in several complex variables An M.S. Thesis Submitted, in partial ulillment o the requirements or the award o the degree o Master o Science in the Faculty o Science, by Chandan Biswas

More information

A dissertation presented. Giulio Tiozzo. The Department of Mathematics

A dissertation presented. Giulio Tiozzo. The Department of Mathematics Entropy, dimension and combinatorial moduli for one-dimensional dynamical systems A dissertation presented by Giulio Tiozzo to The Department of Mathematics in partial fulfillment of the requirements for

More information

CUBIC POLYNOMIAL MAPS WITH PERIODIC CRITICAL ORBIT, PART II: ESCAPE REGIONS

CUBIC POLYNOMIAL MAPS WITH PERIODIC CRITICAL ORBIT, PART II: ESCAPE REGIONS CONFORMAL GEOMETRY AND DYNAMICS An Electronic Journal of the American Mathematical Society Volume 00, Pages 000 000 (Xxxx XX, XXXX) S 1088-4173(XX)0000-0 CUBIC POLYNOMIAL MAPS WITH PERIODIC CRITICAL ORBIT,

More information

1 Differentiable manifolds and smooth maps

1 Differentiable manifolds and smooth maps 1 Differentiable manifolds and smooth maps Last updated: April 14, 2011. 1.1 Examples and definitions Roughly, manifolds are sets where one can introduce coordinates. An n-dimensional manifold is a set

More information

Dynamics of Tangent. Robert L. Devaney Department of Mathematics Boston University Boston, Mass Linda Keen

Dynamics of Tangent. Robert L. Devaney Department of Mathematics Boston University Boston, Mass Linda Keen Dynamics of Tangent Robert L. Devaney Department of Mathematics Boston University Boston, Mass. 02215 Linda Keen Department of Mathematics Herbert H. Lehman College, CUNY Bronx, N.Y. 10468 Abstract We

More information

PERIODS IMPLYING ALMOST ALL PERIODS FOR TREE MAPS. A. M. Blokh. Department of Mathematics, Wesleyan University Middletown, CT , USA

PERIODS IMPLYING ALMOST ALL PERIODS FOR TREE MAPS. A. M. Blokh. Department of Mathematics, Wesleyan University Middletown, CT , USA PERIODS IMPLYING ALMOST ALL PERIODS FOR TREE MAPS A. M. Blokh Department of Mathematics, Wesleyan University Middletown, CT 06459-0128, USA August 1991, revised May 1992 Abstract. Let X be a compact tree,

More information

Connectedness loci of complex polynomials: beyond the Mandelbrot set

Connectedness loci of complex polynomials: beyond the Mandelbrot set Connectedness loci of complex polynomials: beyond the Mandelbrot set Sabyasachi Mukherjee Stony Brook University TIFR, June 2016 Contents 1 Background 2 Antiholomorphic Dynamics 3 Main Theorems (joint

More information

Polynomial Julia sets with positive measure

Polynomial Julia sets with positive measure ? Polynomial Julia sets with positive measure Xavier Buff & Arnaud Chéritat Université Paul Sabatier (Toulouse III) À la mémoire d Adrien Douady 1 / 16 ? At the end of the 1920 s, after the root works

More information

In the index (pages ), reduce all page numbers by 2.

In the index (pages ), reduce all page numbers by 2. Errata or Nilpotence and periodicity in stable homotopy theory (Annals O Mathematics Study No. 28, Princeton University Press, 992) by Douglas C. Ravenel, July 2, 997, edition. Most o these were ound by

More information

Logarithm of a Function, a Well-Posed Inverse Problem

Logarithm of a Function, a Well-Posed Inverse Problem American Journal o Computational Mathematics, 4, 4, -5 Published Online February 4 (http://www.scirp.org/journal/ajcm http://dx.doi.org/.436/ajcm.4.4 Logarithm o a Function, a Well-Posed Inverse Problem

More information

Classification of effective GKM graphs with combinatorial type K 4

Classification of effective GKM graphs with combinatorial type K 4 Classiication o eective GKM graphs with combinatorial type K 4 Shintarô Kuroki Department o Applied Mathematics, Faculty o Science, Okayama Uniervsity o Science, 1-1 Ridai-cho Kita-ku, Okayama 700-0005,

More information

4. Images of Varieties Given a morphism f : X Y of quasi-projective varieties, a basic question might be to ask what is the image of a closed subset

4. Images of Varieties Given a morphism f : X Y of quasi-projective varieties, a basic question might be to ask what is the image of a closed subset 4. Images of Varieties Given a morphism f : X Y of quasi-projective varieties, a basic question might be to ask what is the image of a closed subset Z X. Replacing X by Z we might as well assume that Z

More information

Antipode Preserving Cubic Maps: the Fjord Theorem

Antipode Preserving Cubic Maps: the Fjord Theorem Submitted exclusively to the London Mathematical Society doi:10.1112/0000/000000 Antipode Preserving Cubic Maps: the Fjord Theorem A. Bonifant, X. Buff and John Milnor Abstract This note will study a family

More information

An Algebraic View of the Relation between Largest Common Subtrees and Smallest Common Supertrees

An Algebraic View of the Relation between Largest Common Subtrees and Smallest Common Supertrees An Algebraic View of the Relation between Largest Common Subtrees and Smallest Common Supertrees Francesc Rosselló 1, Gabriel Valiente 2 1 Department of Mathematics and Computer Science, Research Institute

More information

MODULI SPACE OF CUBIC NEWTON MAPS

MODULI SPACE OF CUBIC NEWTON MAPS MODULI SPACE OF CUBIC NEWTON MAPS PASCALE ROESCH, XIAOGUANG WANG, AND YONGCHENG YIN arxiv:1512.05098v2 [math.ds] 18 May 2016 Abstract. In this article, we study the topology and bifurcations of the moduli

More information

ESCAPE QUARTERED THEOREM AND THE CONNECTIVITY OF THE JULIA SETS OF A FAMILY OF RATIONAL MAPS

ESCAPE QUARTERED THEOREM AND THE CONNECTIVITY OF THE JULIA SETS OF A FAMILY OF RATIONAL MAPS ESCAPE QUARTERED THEOREM AND THE CONNECTIVITY OF THE JULIA SETS OF A FAMILY OF RATIONAL MAPS YOU-MING WANG, FEI YANG, SONG ZHANG, AND LIANG-WEN LIAO Abstract. In this article, we investigate the dynamics

More information

EXISTENCE OF QUADRATIC HUBBARD TREES. 1. Introduction

EXISTENCE OF QUADRATIC HUBBARD TREES. 1. Introduction EXISTENCE OF QUADRATIC HUBBARD TREES HENK BRUIN, ALEXANDRA KAFFL, AND DIERK SCHLEICHER Abstract. A (quadratic) Hubbard tree is an invariant tree connecting the critical orbit within the Julia set of a

More information

Rational Maps with Cluster Cycles and the Mating of Polynomials

Rational Maps with Cluster Cycles and the Mating of Polynomials Rational Maps with Cluster Cycles and the Mating of Polynomials Thomas Sharland Institute of Mathematical Sciences Stony Brook University 14th September 2012 Dynamical Systems Seminar Tom Sharland (Stony

More information

GENERALIZED ABSTRACT NONSENSE: CATEGORY THEORY AND ADJUNCTIONS

GENERALIZED ABSTRACT NONSENSE: CATEGORY THEORY AND ADJUNCTIONS GENERALIZED ABSTRACT NONSENSE: CATEGORY THEORY AND ADJUNCTIONS CHRIS HENDERSON Abstract. This paper will move through the basics o category theory, eventually deining natural transormations and adjunctions

More information

Definition We say that a topological manifold X is C p if there is an atlas such that the transition functions are C p.

Definition We say that a topological manifold X is C p if there is an atlas such that the transition functions are C p. 13. Riemann surfaces Definition 13.1. Let X be a topological space. We say that X is a topological manifold, if (1) X is Hausdorff, (2) X is 2nd countable (that is, there is a base for the topology which

More information

EXISTENCE OF ISOPERIMETRIC SETS WITH DENSITIES CONVERGING FROM BELOW ON R N. 1. Introduction

EXISTENCE OF ISOPERIMETRIC SETS WITH DENSITIES CONVERGING FROM BELOW ON R N. 1. Introduction EXISTECE OF ISOPERIMETRIC SETS WITH DESITIES COVERGIG FROM BELOW O R GUIDO DE PHILIPPIS, GIOVAI FRAZIA, AD ALDO PRATELLI Abstract. In this paper, we consider the isoperimetric problem in the space R with

More information

Bordism and the Pontryagin-Thom Theorem

Bordism and the Pontryagin-Thom Theorem Bordism and the Pontryagin-Thom Theorem Richard Wong Differential Topology Term Paper December 2, 2016 1 Introduction Given the classification of low dimensional manifolds up to equivalence relations such

More information

Checkerboard Julia Sets for Rational Maps

Checkerboard Julia Sets for Rational Maps Checkerboard Julia Sets for Rational Maps arxiv:1103.3803v3 [math.ds] 20 Jun 2011 Paul Blanchard Figen Çilingir Daniel Cuzzocreo Robert L. Devaney Daniel M. Look Elizabeth D. Russell Current version: June

More information

Renormalization for Lorenz maps

Renormalization for Lorenz maps Renormalization for Lorenz maps Denis Gaidashev, Matematiska Institutionen, Uppsala Universitet Tieste, June 5, 2012 D. Gaidashev, Uppsala Universitet () Renormalization for Lorenz maps Tieste, June 5,

More information