KLEINIAN GROUPS AND HOLOMORPHIC DYNAMICS
|
|
- Miles Young
- 6 years ago
- Views:
Transcription
1 International Journal of Bifurcation and Chaos, Vol. 3, No. 7 (2003) c World Scientific Publishing Company KLEINIAN GROUPS AND HOLOMORPHIC DYNAMICS C. CORREIA RAMOS Departamento de Matemática and CIMA, Universidade de Évora, Portugal ccr@uevora.pt J. SOUSA RAMOS Departamento de Matemática, Instituto Superior Técnico, Lisboa, Portugal sramos@math.ist.utl.pt Received January 20, 200; Revised December 5, 200 It is known as a correspondence between iteration of rational maps and Kleinian groups, and is usually designated as the Sullivan s dictionary. This dictionary enumerates analogies between iteration of holomorphic endomorphisms and action of Kleinian groups. We propose to study explicitly examples establishing the correspondence between the two theories: iteration theory and Kleinian groups. Keywords: Kleinian groups; holomorphic dynamics; Hecke groups; iteration theory. In this work we explore some relations between iteration theory and theory of Kleinian groups. There is already work done in that direction. In the case of Fuchsian groups there is the result of Bowen and Series [979] which guarantees the existence of a boundary map, orbit equivalent to the action of the Fuchsian group on the hyperbolic plane. We use this result in Sec. 3. Brooks and Matelski [98] used Kleinian groups and complex iteration in relation with the question of deciding when a given twogenerator group is discrete, using the Jorgensen inequality. This approach is also followed by Gehring and Martin [989]. On the other hand, the existence is known of many analogies between iteration theory of holomorphic transformations and the theory of Kleinian groups. These analogies are collected in what is usually called Sullivan s dictionary. The number of these analogies and the importance of the results suggest deep connections between them. We present, in Sec. 2, part of the Sullivan s dictionary. We also present some known results about Kleinian groups which allow us to give explicit examples of relations between iteration theory and Kleinian groups. In Sec. 3 we consider a particular kind of Kleinian groups, the Fuchsian groups, and using the result in [Bowen & Series, 979], we give explicit relations between these groups and the iteration of piecewise fractional linear transformations. We build subshifts associated to Hecke groups, a one-parameter family of Fuchsian groups. In Theorem 3.3 to every Hecke group H q, we associate finite type subshift defined by a transition matrix with dimension q 2. In Theorem 3.4 and Corollary 3.5 we characterize the topological entropy of the subshifts associated to the Hecke groups. The methods here introduced can be generalized to other families of Fuchsian groups. Finally in Sec. 4 we study automorphisms of Kleinian groups and we use it to establish a different kind of relation between iteration theory and Kleinian groups. We focus on a one-parameter holomorphic family of Kleinian groups which 959
2 960 C. Correla Ramos & J. Sousa Ramos corresponds to the family of quasi-fuchsian groups introduced in [Wright, 988]. See also [Keen et al., 993]. Using iteration theory we identify where these groups are in the parameter space. In Theorem 4.2, we state that the parameters for those Kleinian groups do not belong to the Cartesian product of the Mandelbrot set and the filled Julia set of a quadratic map. Thus, a sequence of subgroups defined by iterating the commutator of the generators, have their traces given by a polynomial transformation which implies discreteness criteria for subgroups in P SL 2 (C). -. Preliminaries We can define Kleinian Group as a discrete subgroup Γ of P SL 2 (C). An element of P SL 2 (C) can be represented by a matrix of SL 2 (C) ( ) a b γ = c d Fig.. Limit set of the Kleinian group Γ a with a = 2. with ad bc =, reminding the fact that P SL 2 (C) SL 2 (C)/{±I}. The elements of P SL 2 (C) act on H 3 as orientation-preserving isometries, and act on C as automorphisms, γ(z) = (az + b)/(cz + d), Möbius transformations that preserves orientation. The set Γ(p) = {γ(p) : γ Γ} is denoted as the orbit of p H 3. The set Γ(p) has accumulation points only on the Riemann Sphere C = H 3. These points form a closed set called the limit set of Γ, Λ(Γ) (see Fig., for Γ a = F, G a ( ) ( 0 a =, I a 2 ) a 2, a with a = 2). The set Ω(Γ) = C \Λ(Γ) is called the regular set. According to the type of limit set Λ(Γ), a Kleinian group is called: Elementary, if Λ(Γ) consists of at most two points. Otherwise it is nonelementary. Fuchsian, if Γ has an invariant disc D. The elements in P SL 2 (C) can be classified according to their set of fixed points or, which is equivalent, by the values of the square of the traces for the corresponding matricial representatives: g P SL 2 (C) is parabolic (resp. elliptic, hyperbolic, loxodromic) if T r 2 (g) is 4 (resp. [0, 4[, ]4, + ], / [0, + ]). Fig. 2. Julia set of g β (z) = z( + β z) 2, with β = 2i. Now recall some basic notions on dynamics of rational maps. Given a rational application f : C C we define: The set of escaping points for f E(f) = {z C : lim f n (z) = } n + The filled Julia set K(f) = C E(f).
3 Kleinian Groups and Holomorphic Dynamics 96 The boundary J (f) = K(f) is called the Julia set of f (see Fig. 2). The set M(f) = {c C n : the critical points of f having a bounded orbit under f} is called the Mandelbrot set. 2. Sullivan s Dictionary and Some Known Facts about Kleinian Groups The Sullivan s dictionary consists of analogies between iteration of rational maps and action of Kleinian groups. A more complete description of this dictionary can be found in [Sullivan, 985]. We present here part of it. Complex Dynamics f : C C rational map of degree > 2, Julia set J (f), Fatou set F(f), Mandelbrot set M(f), J (f) : smallest closed nonempty invariant set, #J (f) 3, repelling periodic points are dense in J (f), #{components of F(f)} = 0,, 2, or, No wandering domains theorem: Every f rational with a finite number of singular values has no wandering domain, degree k gives 2k 2 complex parameters. Kleinian Groups Γ Kleinian group, nonelementary and finitely generated, Limit set Λ(Γ), Regular set Ω(Γ), Parameter space Par(Γ), Λ(Γ) : smallest closed nonempty invariant set, #Λ(Γ) 3, loxodromic fixed points are dense in Λ(Γ), #{components of Ω(Γ)} = 0,, 2, or, Ahlfors finiteness theorem: For a finitely generated Kleinian group Γ, Ω(Γ)/Γ is a finite union of analytically finite Riemann Surfaces, n generators give 3n 3 complex parameters. In what follows we present some results that are relevant in the theory of Kleinian groups and which are useful for the objectives of this work. They can be found, for example, in [Beardon, 983]. Concerning the question of deciding if a given group of Möbius transformations is discrete there is an important result on groups with two generators. Theorem 2.. (Jorgensen inequality) If the group generated by A, B P SL 2 (C), A, B, is a nonelementary Kleinian group then Tr 2 (B) 4 + Tr([A, B]) 2. This gives a necessary condition for a given group to be a discrete nonelementary group. As far as we know, there is not a necessary and sufficient condition for a given group to be Kleinian. Nevertheless there is a result that tells us that it is enough to decide the question to groups with two generators. Theorem 2.2. A nonelementary subgroup of P SL 2 (C), Γ, is discrete if and only if each twogenerator subgroup of Γ is discrete. Remark 2.3. There is a recent result by Wang and Yang [2000] which establishes that a nonelementary subgroup of P SL 2 (C), Γ, is discrete if and only if each subgroup of Γ generated by two loxodromic elements is discrete. Concerning the two-generator groups, it is known that a two-generator group Γ = A, B is determined, up to conjugacy by numbers α = Tr 2 (A) 4, β = Tr 2 (B) 4 and γ = Tr([A, B]) 2 provided that γ 0 (see [Gehring et al., 997]). These values are called the parameters of the group Γ. We have then par(γ) = (Tr 2 (A) 4, Tr 2 (B) 4, Tr[A, B] 2) = (α, β, γ) C 3 It would be interesting to describe the subset of C 3 corresponding to the parameters of Kleinian two-generator groups. There is some work done in that direction, see [Gehring et al., 997; Baribeau & Ransford, 2000]. A result that is very useful when trying to prove if a given group is a nonelementary Kleinian group, and which is a step forward in the direction of iteration theory is Proposition 2.4. For A P SL 2 (C) and θ A : P SL 2 (C) P SL 2 (C) θ A (B) = BAB if a Kleinian group A, B satisfies θa n (B) = A for some integer n, and A is not of order 2 then F ix(a) is invariant under B. In particular A, B is elementary.
4 962 C. Correla Ramos & J. Sousa Ramos 3. Relations Between Iteration Theory and Fuchsian Groups 3.. Hecke groups In this section we give an explicit relation between iteration theory and a particular case of Kleinian groups Fuchsian groups. The possibility of associating to a Fuchsian group a map is known, defined on the boundary of H 2, R { }, that is orbit equivalent to the group [Bowen & Series, 979]. We build explicitly this map for the Hecke groups. A Hecke group, H q, is a one-parameter family of Fuchsian groups generated by the following transformations A(z) = u( z) and B(z) = z. If u < 4, the groups are discrete only for the parameter of the form u = 4 cos 2 (π/q), q integer q 3 (the case q = 3 corresponds to the modular group P SL 2 (Z)). The boundary map is built taking into account an appropriate fundamental domain for the group. There are many possible choices. We have chosen as the fundamental domain for the Hecke groups the polygon delimited by the vertices i, i+, (+i 3)/2,. In fact we build a boundary map for each group H q, so we have a one-parameter family of maps, f q. The family of piecewise fractional-linear transformations associated to the Hecke groups, are q= Fig. 3. Graph of f q when q = 6, and the orbit of the point 0. given by B (x) if x A (x) if < x 0 f q (x) = A (x) if 0 < x B (x) if x For each map, f q, we can build a Markov partition and the associated Markov matrix, M q. Example 3.. For the case q = 3 we have the following partition I =], ], I 2 =], 0], I 3 =]0, ] I 4 =], 2], I 42 =]2, + ] and the matrix M 3 = Example 3.2. case q = 4 We present here the matrix for the M 4 = In general we have the following result Theorem 3.3. To every Hecke group H q we can associate the Markov matrix M q = with dim(m q ) = q + 2.
5 Kleinian Groups and Holomorphic Dynamics 963 Proof. The partition for the general case is given by the following with I = [, ] I 2 = [, 0] I 3 = [0, x ] I 32 = [x 2, x 3 ] q 2 I I 2 I 3(q 2) = [x q 3, ] I 4 = [, + x ] I 42 = [ + x, + ] i=2 I 3i I 4 I 42 = R { } and where x = /u, and x i = A(x i ), A being one of the group generators, defined above. Looking for the orbits of the critical points we can determine the possible transitions by the action of the boundary map, f q. We have f q (I ) = I I 2 f q (I 2 ) = I 42 f q (I 3i ) = I 3(i+), i =,..., q 3 f q (I 3(q 2) ) = I 4 I 42 f q (I 4 ) = I 3 f q (I 42 ) = q 2 i=2 I 3i I 4 I 42 So the matrix, that codifies these transitions is M q presented in the statement. With matrices, it is possible to determine the topological entropy that becomes an invariant for the Hecke group via topological dynamics. The topological entropy may be determined as the logarithm of the growth number, s(f q ), s = lim n n l(fq n ) where l(f n q ) is the lap number of f n q. It is known to be equal to the spectral radius of the matrix M q (see [Lind & Marcus, 995]). We present some of the growth numbers s for the groups associated with the first values of q: q s (aprox.) It is clear that the growth numbers approach the value 2. In fact we prove the following results Theorem 3.4. The topological entropy of the subshift associated to the Hecke groups H q is given by h t = log s, where s (spectral radius of the matrix M q ) is the greatest real zero of the polynomial p q (t) = t q t q 2 t. Proof. Consider the matrix M q ti q+2 where I q+2 is the (q + 2) (q + 2) identity matrix. This matrix is reducible and is decomposed in the following way t 0 t M q ti q+2 = 0 0 b t b,q b q, b q,q t so det(m q ti q+2 ) = t(t ) det(b q ti q ). If we set N q ti q = P (B q ti q )Q, where P, Q are convenient elementary matrices, we obtain a matrix N q ( ) Cq ti q 0 N q ti q = t such that det(n q ti q ) = det(b q ti q ) and C q is the companion matrix with characteristic polynomial t q t q 2 t. Corollary 3.5. The entropies h t, associated to the subshifts M q, for the Hecke groups H q, are bounded above by log 2. Proof. Relating the polynomial p q (t) = t q t q 2 t, as q, with the geometric series, it is easy to see that the zero of p (t) is t = 2. It is relevant to note that like in the Kneading Theory [Milnor & Thurston, 988] the dynamics here is completely determined by the symbolic sequences (kneading sequences) corresponding to the
6 964 C. Correla Ramos & J. Sousa Ramos itineraries of the lateral limits of the discontinuous points. In this context the symbolic sequences have additional meaning than just symbolic coding, they have a direct interpretation in group words. So we may speak of kneading words in the group, and they codify topological information about the group. We have the following kneading words discontinuous point kneading word + A BA q 3 B[A q 2 B] + B A [B] 0 + [A q 2 B] 0 + A [B] + B[A q 2 B] + A[B] 3.2. Fuchsian groups generated by two parabolic transformations We now consider a different family of Fuchsian groups, those generated by the parabolic transformations A ρ (z) = z ρz +, and B ρ(z) = z + ρ where ρ R. We set G ρ = A ρ, B ρ. This group is discrete if and only if ρ = 2 cos(π/q), q 3 or ρ 2 (see [Beardon, 988]). Based on the fundamental domain determined by the region Re(z) < ρ/2, and the exterior of the circles z (/ρ) = /ρ, z + (/ρ) = /ρ, we have the following boundary map, for the case ρ =, B (x) if x 2 A (x) if 2 < x 0 f(x) = A (x) if 0 < x 2 (x) if x 2 B The Markov matrix is M = To exemplify how this construction carries topological information, we present other family of Fuchsian groups, in which if we change continuously the parameters, the partition is always the same, and so the topological information is the same. The surface obtained by H 2 /G is the independent of the parameters values. The generators are with A(z) = λ 2 z, and B(z) = pz + p2 r 2 z + p r = p λ2 λ 2 +, p, λ R. The family of associated boundary maps is the following A (x) if x p r B (x) if p r < x p + r f λ,p (x) = A(x) if p + r < x p r B(x) if p r < x p + r A (x) if p + r < x The partition and consequently the Markov matrix M = are independent of the parameters λ, p. The topological entropy associated with the Markov subshift is log 3, which indicates the fact that the group is free. See in Fig. 4 a fundamental domain and Fig. 5 the map f 2, Fig. 4. Fundamental domain for G λ,p = A λ,p, B λ,p, λ = 2 and p =.
7 Kleinian Groups and Holomorphic Dynamics Fig. 5. Graph of the map f 2,. 4. Some Special Automorphisms of the Kleinian Groups As was said before, given a group generated by two transformations, it is possible to assign parameters that characterize the group up to conjugacies (see [Gehring et al., 997]). We write par(γ) = (Tr 2 (A) 4, Tr 2 (B) 4, Tr[A, B] 2) = (α, β, γ) C 3 With this notation the Jorgensen inequality is expressed by β + γ. We have that if β + γ < then the group Γ is either not discrete or elementary. It is possible to explore some of the twogenerator subgroups of Γ using automorphisms in the group. If we consider the following automorphism θ B : P SL 2 (C) P SL 2 (C) θ B (A) = ABA it is possible to prove that, defining we have z n = Tr[B, θ n B A] 2, z n = z n (z n β). If we iterate f β (z) = z(z β), with the initial condition z 0 = γ, we are obtaining the subgroups Γ n with par(γ n ) = (β, β, z n ) = (β, β, z n (z n β)). Conclusions on discreteness of the group Γ can be taken from this iteration since if we know that some subgroup is not discrete then the group itself cannot be discrete. In [Brooks & Matelski, 98] and [Gehring & Martin, 989] the authors identify the Jorgensen inequality as the basin attraction of the fixed point. We consider the set (M(f β ) K(f β )), the interior of the Cartesian product of the Mandelbrot set, M(f β ) and the filled Julia set K(f β ) for the map f β (z). Elements (β, γ) (M(f β ) K(f β )) correspond to nondiscrete or elementary groups Γ with par(γ) = (α, β, γ). With the automorphism θ B only a particular path in the group is explored. It is possible to give different kinds of automorphisms with different associated polynomials. Iteration with group automorphisms corresponds to iteration of polynomials with the parameter γ as initial condition. Another example is φ B : P SL 2 (C) P SL 2 (C) φ B (A) = ABA BA The polynomial associated to φ B is g β (z) = z( + β z) 2, this time is a cubic (see Fig. 2 for an example of Julia set). The sequence of subgroups is now the following par(γ n ) = (α n, β, z n ( + β z n ) 2 ). For the automorphism φ B we have the set (M(g β ) K(g β )) associated to the map g β (z). In general we have the following relations ψ B p β (M(p β ) K(p β )) where ψ B is an group automorphism, p β is the associated polynomial, and M(p β ), K(p β ) the corresponding Mandelbrot and filled Julia set. In what follows, we consider the one-parameter family of Kleinian groups G ξ = A, B ξ, with ξ C, where the generator representatives in SL 2 (C) are given by A = ( ) 2 0 and B ξ = ( iξ ) i i 0 Note that Tr[A, B] = 2. We are going to identify where these groups are in the parameter space, i.e. we are looking for par(g ξ ). Such groups were introduced in [Wright, 988] and [Keen et al., 993] where the authors identify the cusp groups. For each ξ C such Tr(W p/q ) = ±2, G ξ corresponds to a cusp group. The words W p/q are built recursively by the following process W /0 = A W / = AB ξ W / = A B ξ W 0/ = B ξ W (p+r)/(q+s) = W p/q W r/s
8 966 C. Correla Ramos & J. Sousa Ramos with p q = + r/s > r/s. The traces of this group words are polynomials in the variable ξ C. The recursion on the words, and the trace identity Tr(XY ) = Tr(X)Tr(Y ) Tr(XY ) for any matrices X, Y SL 2 (C) allow a definition of an iterative procedure to explore paths in the group to determine for which ξ corresponds to a cusp group. Let c = Tr(Y ), t 0 = Tr(X) and t = Tr(XY ), we have t n+2 = ct n+ t n where t n = Tr(XY n ), X = W p/q and Y = W r/s for some rationales p/q, r/s. Remark ]. For the trace identities see [Magnus, The parameter c depends on ξ. There is a correspondence between the set of cusp groups and the set of parameters c C, namely the values of ξ C, for which there is n N such that t n = ±2. If the condition < Im(ξ) < 2 is held, then the corresponding group is discrete. We have the following result which states that for this family the parameters are outside both (M(f β ) K(f β )) and (M(g β ) K(g β )). Theorem 4.2. Let G ξ = A, B ξ be the oneparameter family of Kleinian groups defined above. If then and par(g ξ ) = (α, β(ξ), γ) = (0, ξ 2 4, 4) (β(ξ), 4) / (M(f β ) K(f β )) (β(ξ), 4) / (M(g β ) K(g β )). Proof. First we prove the result for the map f β. Since (β(ξ), 4) C R is sufficient to prove that γ = 4 / Re K o (f β ), β. Let β R, β 0. For K o (f β ) to be connected we must have that the image, by f β, of the critical point β/2 is greater than the preimage of the fixed point β +. This condition gives (β 2 /4), which is equivalent to β 2. If we set β R, β < 0, we get a similar condition in order for K o (f β ) to be connected. In this case the condition is that the image of the critical point be greater than the preimage of the fixed point 0, which is equivalent to β 4. Then the point ( 4, 4) belongs to (M(f β ) K(f β )), and for values of β 4, we have γ 4, proving the result. For the map g β with the same arguments we see that M(f β ) R = [ 4, 2], and that the line (β, 4), intersects M(f β ) K(f β ) only on ( 4, 4), and the result follows. It is natural to think in a generalization of other group automorphisms. Let G ξ = A, B ξ as above. Let ψ B be an automorphism in G with associated polynomial p β. Then (β(ξ), 4) / (M(p β ) K(p β )). Acknowledgment The authors would like to thank the referee for his suggestions and advice in the presentation of this paper. References Baribeau, L. & Ransford, T. [2000] On the set of discrete two-generator groups, Math. Proc. Camb. Phil. Soc. 28, Beardon, A. [983] The Geometry of Discrete Groups (Springer-Verlag). Beardon, A. [988] Fuchsian groups and nth roots of parabolic generators, Holomorphic Functions and Moduli II, Mathematical Sciences Research Institute Publications (Springer-Verlag), pp Beardon, A. [993] Pell s equation and two generator free Möbius groups, Bull. London Math. Soc. 25, Bowen, R. & Series, C. [979] Markov maps associated with Fuchsian groups, I.H.E.S. Publications 50, Brooks, R. & Matelski, P. [98] The dynamics of 2-generator subgroups of P SL 2 (C), Riemann Surfaces and Related Topics, Proc. 978 Stony Brook Conf., Annals of Mathematical Studies (Princeton University Press). Gehring, F. & Martin, G. [989] Iteration theory and inequalities for Kleinian groups, Bull. Amer. Math. Soc. 2, Gehring, F. W., Maclachlan, G., Martin, G. & Reid, A. W. [997] Arithmeticity, discreteness and volume, Trans. Amer. Math. Soc. 349, Keen, L. & Series, C. [993] Pleating coordinates for Maskit embedding of the Teichmüller space of punctured tori, Topology 32, Lind, D. & Marcus, B. [995] An Introduction to Symbolic Dynamics and Coding (Cambridge University Press).
9 Kleinian Groups and Holomorphic Dynamics 967 Magnus, W. [974] Non Euclidean Tessellations and Their Groups (Academic Press). Milnor, J. & Thurston, W. [988] On iterated maps of the interval, Proc. Univ. Maryland , ed. Alexander, J. C., Lecture Notes in Mathematics, Vol. 342 (Springer-Verlag), pp Series, C. [999] Lectures on pleating coordinates for once punctured tori, Warwick preprint 07/999. Sullivan, D. [985] Quasiconformal homeomorphisms and dynamics I. Solution of the Fatou Julia problem on wandering domains, Ann. Math. 22, Wright, D. [988] The shape of the boundary of Maskit s embedding of Teichmüller space of once punctured tori, unpublished preprint (see [Series, 999]).
10
On the local connectivity of limit sets of Kleinian groups
On the local connectivity of limit sets of Kleinian groups James W. Anderson and Bernard Maskit Department of Mathematics, Rice University, Houston, TX 77251 Department of Mathematics, SUNY at Stony Brook,
More informationMöbius transformations Möbius transformations are simply the degree one rational maps of C: cz + d : C C. ad bc 0. a b. A = c d
Möbius transformations Möbius transformations are simply the degree one rational maps of C: where and Then σ A : z az + b cz + d : C C ad bc 0 ( ) a b A = c d A σ A : GL(2C) {Mobius transformations } is
More informationRANDOM HOLOMORPHIC ITERATIONS AND DEGENERATE SUBDOMAINS OF THE UNIT DISK
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 00, Number 0, Pages 000 000 S 0002-9939(XX)0000-0 RANDOM HOLOMORPHIC ITERATIONS AND DEGENERATE SUBDOMAINS OF THE UNIT DISK LINDA KEEN AND NIKOLA
More informationSOME REMARKS ON NON-DISCRETE MÖBIUS GROUPS
Annales Academiæ Scientiarum Fennicæ Mathematica Volumen 21, 1996, 69 79 SOME REMARKS ON NON-DISCRETE MÖBIUS GROUPS A. F. Beardon University of Cambridge, Department of Pure Mathematics and Mathematical
More information676 JAMES W. ANDERSON
676 JAMES W. ANDERSON Such a limit set intersection theorem has been shown to hold under various hypotheses. Maskit [17] shows that it holds for pairs of analytically finite component subgroups of a Kleinian
More informationCOMPLETELY INVARIANT JULIA SETS OF POLYNOMIAL SEMIGROUPS
Series Logo Volume 00, Number 00, Xxxx 19xx COMPLETELY INVARIANT JULIA SETS OF POLYNOMIAL SEMIGROUPS RICH STANKEWITZ Abstract. Let G be a semigroup of rational functions of degree at least two, under composition
More informationSystoles of hyperbolic 3-manifolds
Math. Proc. Camb. Phil. Soc. (2000), 128, 103 Printed in the United Kingdom 2000 Cambridge Philosophical Society 103 Systoles of hyperbolic 3-manifolds BY COLIN C. ADAMS Department of Mathematics, Williams
More informationSingular 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 informationarxiv: v2 [math.gt] 16 Jan 2009
CONJUGATE GENERATORS OF KNOT AND LINK GROUPS arxiv:0806.3452v2 [math.gt] 16 Jan 2009 JASON CALLAHAN Abstract. This note shows that if two elements of equal trace (e.g., conjugate elements generate an arithmetic
More informationEvolution 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 informationSOME SPECIAL 2-GENERATOR KLEINIAN GROUPS
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 106, Number I, May 1989 SOME SPECIAL 2-GENERATOR KLEINIAN GROUPS BERNARD MASKIT (Communicated by Irwin Kra) Abstract. We explore the following question.
More informationTHE FUNDAMENTAL GROUP OF THE DOUBLE OF THE FIGURE-EIGHT KNOT EXTERIOR IS GFERF
THE FUNDAMENTAL GROUP OF THE DOUBLE OF THE FIGURE-EIGHT KNOT EXTERIOR IS GFERF D. D. LONG and A. W. REID Abstract We prove that the fundamental group of the double of the figure-eight knot exterior admits
More informationThe 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 informationAction of mapping class group on extended Bers slice
Action of mapping class group on extended Bers slice (Kentaro Ito) 1 Introduction Let S be an oriented closed surface of genus g 2. Put V (S) = Hom(π 1 (S), PSL 2 (C))/PSL 2 (C). Let X be an element of
More informationPart II. Geometry and Groups. Year
Part II Year 2014 2013 2012 2011 2010 2009 2008 2007 2006 2005 2014 Paper 4, Section I 3F 49 Define the limit set Λ(G) of a Kleinian group G. Assuming that G has no finite orbit in H 3 S 2, and that Λ(G),
More informationSOME SPECIAL KLEINIAN GROUPS AND THEIR ORBIFOLDS
Proyecciones Vol. 21, N o 1, pp. 21-50, May 2002. Universidad Católica del Norte Antofagasta - Chile SOME SPECIAL KLEINIAN GROUPS AND THEIR ORBIFOLDS RUBÉN HIDALGO Universidad Técnica Federico Santa María
More informationKleinian Groups with Real Parameters
Kleinian Groups with Real Parameters FW Gehring, J P Gilman and GJ Martin Dedicated to Albert Marden on the occasion of his 65 th birthday Abstract We find all real points of the analytic space of two
More informationFixed Points & Fatou Components
Definitions 1-3 are from [3]. Definition 1 - A sequence of functions {f n } n, f n : A B is said to diverge locally uniformly from B if for every compact K A A and K B B, there is an n 0 such that f n
More informationZoology of Fatou sets
Math 207 - Spring 17 - François Monard 1 Lecture 20 - Introduction to complex dynamics - 3/3: Mandelbrot and friends Outline: Recall critical points and behavior of functions nearby. Motivate the proof
More informationOn 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 informationSiegel Discs in Complex Dynamics
Siegel Discs in Complex Dynamics Tarakanta Nayak, Research Scholar Department of Mathematics, IIT Guwahati Email: tarakanta@iitg.ernet.in 1 Introduction and Definitions A dynamical system is a physical
More informationON THE MARGULIS CONSTANT FOR KLEINIAN GROUPS, I
Annales Academiæ Scientiarum Fennicæ Mathematica Volumen 21, 1996, 439 462 ON THE MARGULIS CONSTANT FOR KLEINIAN GROUPS, I F.W. Gehring and G.J. Martin University of Michigan, Department of Mathematics
More informationEquivalent trace sets for arithmetic Fuchsian groups
Equivalent trace sets for arithmetic Fuchsian groups Grant S Lakeland December 30 013 Abstract We show that the modular group has an infinite family of finite index subgroups each of which has the same
More informationABELIAN COVERINGS, POINCARÉ EXPONENT OF CONVERGENCE AND HOLOMORPHIC DEFORMATIONS
Annales Academiæ Scientiarum Fennicæ Series A. I. Mathematica Volumen 20, 1995, 81 86 ABELIAN COVERINGS, POINCARÉ EXPONENT OF CONVERGENCE AND HOLOMORPHIC DEFORMATIONS K. Astala and M. Zinsmeister University
More informationTOPOLOGICAL DYNAMICS OF DIANALYTIC MAPS ON KLEIN SURFACES
Submitted to Topology Proceedings TOPOLOGICAL DYNAMICS OF DIANALYTIC MAPS ON KLEIN SURFACES JANE HAWKINS Abstract. We study analytic maps of the sphere, the torus and the punctured plane with extra symmetries;
More informationCommensurability between once-punctured torus groups and once-punctured Klein bottle groups
Hiroshima Math. J. 00 (0000), 1 34 Commensurability between once-punctured torus groups and once-punctured Klein bottle groups Mikio Furokawa (Received Xxx 00, 0000) Abstract. The once-punctured torus
More informationArtin-Mazur Zeta Function on Trees with Infinite Edges
5 10 July 2004, Antalya, Turkey Dynamical Systems and Applications, Proceedings, pp. 65 73 Artin-Mazur Zeta Function on Trees with Infinite Edges J. F. Alves and J. Sousa Ramos Departamento de Matemática,
More informationQUASICONFORMAL HOMOGENEITY OF HYPERBOLIC MANIFOLDS
QUASICONFORMAL HOMOGENEITY OF HYPERBOLIC MANIFOLDS PETRA BONFERT-TAYLOR, RICHARD D. CANARY, GAVEN MARTIN, AND EDWARD TAYLOR Abstract. We exhibit strong constraints on the geometry and topology of a uniformly
More informationELEMENTARY PROOF OF THE CONTINUITY OF THE TOPOLOGICAL ENTROPY AT θ = 1001 IN THE MILNOR THURSTON WORLD
REVISTA DE LA UNIÓN MATEMÁTICA ARGENTINA Vol. 58, No., 207, Pages 63 88 Published online: June 3, 207 ELEMENTARY PROOF OF THE CONTINUITY OF THE TOPOLOGICAL ENTROPY AT θ = 00 IN THE MILNOR THURSTON WORLD
More informationTrace fields of knots
JT Lyczak, February 2016 Trace fields of knots These are the knotes from the seminar on knot theory in Leiden in the spring of 2016 The website and all the knotes for this seminar can be found at http://pubmathleidenunivnl/
More informationHunting Admissible Kneading pairs of a Real Rational Map
Hunting Admissible Kneading pairs of a Real Rational Map João Cabral 1 1 Department of Mathematics of University of Azores Ponta Delgada, Portugal ABSTRACT The importance of symbolic systems is that they
More informationIn 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 informationFuchsian groups. 2.1 Definitions and discreteness
2 Fuchsian groups In the previous chapter we introduced and studied the elements of Mob(H), which are the real Moebius transformations. In this chapter we focus the attention of special subgroups of this
More informationPeriodic cycles and singular values of entire transcendental functions
Periodic cycles and singular values of entire transcendental functions Anna Miriam Benini and Núria Fagella Universitat de Barcelona Barcelona Graduate School of Mathematics CAFT 2018 Heraklion, 4th of
More informationOn the Margulis constant for Kleinian groups, I
arxiv:math/9504209v1 [math.dg] 7 Apr 1995 On the Margulis constant for Kleinian groups, I F. W. Gehring G. J. Martin May 1, 2011 Abstract The Margulis constant for Kleinian groups is the smallest constant
More informationPRELIMINARY LECTURES ON KLEINIAN GROUPS
PRELIMINARY LECTURES ON KLEINIAN GROUPS KEN ICHI OHSHIKA In this two lectures, I shall present some backgrounds on Kleinian group theory, which, I hope, would be useful for understanding more advanced
More informationThe bumping set and the characteristic submanifold
The bumping set and the characteristic submanifold Abstract We show here that the Nielsen core of the bumping set of the domain of discontinuity of a Kleinian group Γ is the boundary for the characteristic
More informationCHAOTIC UNIMODAL AND BIMODAL MAPS
CHAOTIC UNIMODAL AND BIMODAL MAPS FRED SHULTZ Abstract. We describe up to conjugacy all unimodal and bimodal maps that are chaotic, by giving necessary and sufficient conditions for unimodal and bimodal
More informationThis article appeared in a journal published by Elsevier. The attached copy is furnished to the author for internal non-commercial research and
This article appeared in a journal published by Elsevier. The attached copy is furnished to the author for internal non-commercial research and education use, including for instruction at the authors institution
More informationDiscrete groups and the thick thin decomposition
CHAPTER 5 Discrete groups and the thick thin decomposition Suppose we have a complete hyperbolic structure on an orientable 3-manifold. Then the developing map D : M H 3 is a covering map, by theorem 3.19.
More informationHyperbolic 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 informationDynamics 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 informationDISCRETENESS CRITERIA FOR ISOMETRY GROUPS OF NEGATIVE CURVATURE. Binlin Dai and Bing Nai. Received April 3, 2003; revised September 5, 2003
Scientiae Mathematicae Japonicae Online, Vol. 9, (2003), 411 417 411 DISCRETENESS CRITERIA FOR ISOMETRY GROUPS OF NEGATIVE CURVATURE Binlin Dai and Bing Nai Received April 3, 2003; revised September 5,
More informationarxiv:math/ v1 [math.gt] 15 Aug 2003
arxiv:math/0308147v1 [math.gt] 15 Aug 2003 CIRCLE PACKINGS ON SURFACES WITH PROJECTIVE STRUCTURES AND UNIFORMIZATION SADAYOSHI KOJIMA, SHIGERU MIZUSHIMA, AND SER PEOW TAN Abstract. Let Σ g be a closed
More informationTHE VOLUME OF A HYPERBOLIC 3-MANIFOLD WITH BETTI NUMBER 2. Marc Culler and Peter B. Shalen. University of Illinois at Chicago
THE VOLUME OF A HYPERBOLIC -MANIFOLD WITH BETTI NUMBER 2 Marc Culler and Peter B. Shalen University of Illinois at Chicago Abstract. If M is a closed orientable hyperbolic -manifold with first Betti number
More informationOn the topology of H(2)
On the topology of H(2) Duc-Manh Nguyen Max-Planck-Institut für Mathematik Bonn, Germany July 19, 2010 Translation surface Definition Translation surface is a flat surface with conical singularities such
More information10. Classifying Möbius transformations: conjugacy, trace, and applications to parabolic transformations
10. Classifying Möbius transformations: conjugacy, trace, and applications to parabolic transformations 10.1 Conjugacy of Möbius transformations Before we start discussing the geometry and classification
More informationRUBÉN A. HIDALGO Universidad Técnica Federico Santa María - Chile
Proyecciones Vol 19 N o pp 157-196 August 000 Universidad Católica del Norte Antofagasta - Chile FIXED POINT PARAMETERS FOR MÖBIUS GROUPS RUBÉN A HIDALGO Universidad Técnica Federico Santa María - Chile
More informationStream lines, quasilines and holomorphic motions
arxiv:1407.1561v1 [math.cv] 7 Jul 014 Stream lines, quasilines and holomorphic motions Gaven J. Martin Abstract We give a new application of the theory of holomorphic motions to the study the distortion
More informationVisualizing the unit ball for the Teichmüller metric
Visualizing the unit ball for the Teichmüller metric Ronen E. Mukamel October 9, 2014 Abstract We describe a method to compute the norm on the cotangent space to the moduli space of Riemann surfaces associated
More informationON THE NIELSEN-THURSTON-BERS TYPE OF SOME SELF-MAPS OF RIEMANN SURFACES WITH TWO SPECIFIED POINTS
Imayoshi, Y., Ito, M. and Yamamoto, H. Osaka J. Math. 40 (003), 659 685 ON THE NIELSEN-THURSTON-BERS TYPE OF SOME SELF-MAPS OF RIEMANN SURFACES WITH TWO SPECIFIED POINTS Dedicated to Professor Hiroki Sato
More informationHighly complex: Möbius transformations, hyperbolic tessellations and pearl fractals
Highly complex: Möbius transformations, hyperbolic tessellations and pearl fractals Department of mathematical sciences Aalborg University Cergy-Pontoise 26.5.2011 Möbius transformations Definition Möbius
More informationDYNAMICS OF RATIONAL SEMIGROUPS
DYNAMICS OF RATIONAL SEMIGROUPS DAVID BOYD AND RICH STANKEWITZ Contents 1. Introduction 2 1.1. The expanding property of the Julia set 4 2. Uniformly Perfect Sets 7 2.1. Logarithmic capacity 9 2.2. Julia
More informationREVERSALS ON SFT S. 1. Introduction and preliminaries
Trends in Mathematics Information Center for Mathematical Sciences Volume 7, Number 2, December, 2004, Pages 119 125 REVERSALS ON SFT S JUNGSEOB LEE Abstract. Reversals of topological dynamical systems
More informationDISCRETENESS CRITERIA AND THE HYPERBOLIC GEOMETRY OF PALINDROMES
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 DISCRETENESS CRITERIA AND THE HYPERBOLIC GEOMETRY
More informationDISCRETENESS CRITERIA AND THE HYPERBOLIC GEOMETRY OF PALINDROMES
DISCRETENESS CRITERIA AND THE HYPERBOLIC GEOMETRY OF PALINDROMES JANE GILMAN AND LINDA KEEN Abstract. We consider non-elementary representations of two generator free groups in P SL(2, C), not necessarily
More informationTHE SLICE DETERMINED BY MODULI EQUATION xy=2z IN THE DEFORMATION SPACE OF ONCE PUNCTURED TORI
Sasaki, T. Osaka J. Math. 33 (1996), 475-484 THE SLICE DETERMINED BY MODULI EQUATION xy=2z IN THE DEFORMATION SPACE OF ONCE PUNCTURED TORI TAKEHIKO SASAKI (Received February 9, 1995) 1. Introduction and
More informationarxiv: 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 informationSOME COUNTEREXAMPLES IN DYNAMICS OF RATIONAL SEMIGROUPS. 1. Introduction
SOME COUNTEREXAMPLES IN DYNAMICS OF RATIONAL SEMIGROUPS RICH STANKEWITZ, TOSHIYUKI SUGAWA, AND HIROKI SUMI Abstract. We give an example of two rational functions with non-equal Julia sets that generate
More informationGeometry of Hyperbolic Components in the Interior of the Mandelbrot Set
Geometry of Hyperbolic Components in the Interior of the Mandelbrot Set Erez Nesharim Written under the supervision of professor Genadi Levin Submitted as a Master s Thesis at the Einstein Institute of
More informationb 0 + b 1 z b d z d
I. Introduction Definition 1. For z C, a rational function of degree d is any with a d, b d not both equal to 0. R(z) = P (z) Q(z) = a 0 + a 1 z +... + a d z d b 0 + b 1 z +... + b d z d It is exactly
More informationTHE CONLEY ATTRACTORS OF AN ITERATED FUNCTION SYSTEM
Bull. Aust. Math. Soc. 88 (2013), 267 279 doi:10.1017/s0004972713000348 THE CONLEY ATTRACTORS OF AN ITERATED FUNCTION SYSTEM MICHAEL F. BARNSLEY and ANDREW VINCE (Received 15 August 2012; accepted 21 February
More informationUniversität Dortmund, Institut für Mathematik, D Dortmund (
Jordan and Julia Norbert Steinmetz Universität Dortmund, Institut für Mathematik, D 44221 Dortmund (e-mail: stein@math.uni-dortmund.de) Received: 8 November 1995 / Revised version: Mathematics Subject
More informationAntipode 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 informationAccumulation constants of iterated function systems with Bloch target domains
Accumulation constants of iterated function systems with Bloch target domains September 29, 2005 1 Introduction Linda Keen and Nikola Lakic 1 Suppose that we are given a random sequence of holomorphic
More informationRational 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 informationCaroline Series Publications
Caroline Series Publications Mathematics Institute, Warwick University, Coventry CV4 7AL, UK C.M.Series@warwick.ac.uk http:www.maths.warwick.ac.uk/ masbb/ Research Papers 1. Ergodic actions of product
More informationINVISIBLE 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 informationIntroduction to Arithmetic Geometry Fall 2013 Lecture #23 11/26/2013
18.782 Introduction to Arithmetic Geometry Fall 2013 Lecture #23 11/26/2013 As usual, a curve is a smooth projective (geometrically irreducible) variety of dimension one and k is a perfect field. 23.1
More informationTHE THERMODYNAMIC FORMALISM APPROACH TO SELBERG'S ZETA FUNCTION FOR PSL(2, Z)
BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY Volume 25, Number 1, July 1991 THE THERMODYNAMIC FORMALISM APPROACH TO SELBERG'S ZETA FUNCTION FOR PSL(2, Z) DIETER H. MAYER I. INTRODUCTION Besides
More informationDECOMPOSING DIFFEOMORPHISMS OF THE SPHERE. inf{d Y (f(x), f(y)) : d X (x, y) r} H
DECOMPOSING DIFFEOMORPHISMS OF THE SPHERE ALASTAIR FLETCHER, VLADIMIR MARKOVIC 1. Introduction 1.1. Background. A bi-lipschitz homeomorphism f : X Y between metric spaces is a mapping f such that f and
More informationCOMPLEX ANALYSIS-II HOMEWORK
COMPLEX ANALYSIS-II HOMEWORK M. LYUBICH Homework (due by Thu Sep 7). Cross-ratios and symmetries of the four-punctured spheres The cross-ratio of four distinct (ordered) points (z,z 2,z 3,z 4 ) Ĉ4 on the
More informationS. Morosawa, Y. Nishimura, M. Taniguchi, T. Ueda. Kochi University, Osaka Medical College, Kyoto University, Kyoto University. Holomorphic Dynamics
S. Morosawa, Y. Nishimura, M. Taniguchi, T. Ueda Kochi University, Osaka Medical College, Kyoto University, Kyoto University Holomorphic Dynamics PUBLISHED BY THE PRESS SYNDICATE OF THE UNIVERSITY OF CAMBRIDGE
More informationAPPLICATION OF HOARE S THEOREM TO SYMMETRIES OF RIEMANN SURFACES
Annales Academiæ Scientiarum Fennicæ Series A. I. Mathematica Volumen 18, 1993, 307 3 APPLICATION OF HOARE S THEOREM TO SYMMETRIES OF RIEMANN SURFACES E. Bujalance, A.F. Costa, and D. Singerman Universidad
More informationThe Geometry of Two Generator Groups: Hyperelliptic Handlebodies
The Geometry of Two Generator Groups: Hyperelliptic Handlebodies Jane Gilman (gilman@andromeda.rutgers.edu) Department of Mathematics, Smith Hall Rutgers University, Newark, NJ 07102, USA Linda Keen (linda.keen@lehman.cuny.edu)
More informationA FUCHSIAN GROUP PROOF OF THE HYPERELLIPTICITY OF RIEMANN SURFACES OF GENUS 2
Annales Academiæ Scientiarum Fennicæ Mathematica Volumen 28, 2003, 69 74 A FUCHSIAN GROUP PROOF OF THE HYPERELLIPTICITY OF RIEMANN SURFACES OF GENUS 2 Yolanda Fuertes and Gabino González-Diez Universidad
More informationBjorn Poonen. Cantrell Lecture 3 University of Georgia March 28, 2008
University of California at Berkeley Cantrell Lecture 3 University of Georgia March 28, 2008 Word Isomorphism Can you tile the entire plane with copies of the following? Rules: Tiles may not be rotated
More informationarxiv: v2 [math.ds] 9 Jun 2013
SHAPES OF POLYNOMIAL JULIA SETS KATHRYN A. LINDSEY arxiv:209.043v2 [math.ds] 9 Jun 203 Abstract. Any Jordan curve in the complex plane can be approximated arbitrarily well in the Hausdorff topology by
More informationBending deformation of quasi-fuchsian groups
Bending deformation of quasi-fuchsian groups Yuichi Kabaya (Osaka University) Meiji University, 30 Nov 2013 1 Outline The shape of the set of discrete faithful representations in the character variety
More informationChapter 6: The metric space M(G) and normal families
Chapter 6: The metric space MG) and normal families Course 414, 003 04 March 9, 004 Remark 6.1 For G C open, we recall the notation MG) for the set algebra) of all meromorphic functions on G. We now consider
More informationThe Geometry of Two Generator Groups: Hyperelliptic Handlebodies
The Geometry of Two Generator Groups: Hyperelliptic Handlebodies Jand Gilman (gilman@andromeda.rutgers.edu) Department of Mathematics, Smith Hall Rutgers University, Newark, NJ 07102, USA Linda Keen (linda@lehman.cuny.edu)
More informationINVARIANTS OF TWIST-WISE FLOW EQUIVALENCE
ELECTRONIC RESEARCH ANNOUNCEMENTS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 3, Pages 126 130 (December 17, 1997) S 1079-6762(97)00037-1 INVARIANTS OF TWIST-WISE FLOW EQUIVALENCE MICHAEL C. SULLIVAN (Communicated
More informationMinicourse on Complex Hénon Maps
Minicourse on Complex Hénon Maps (joint with Misha Lyubich) Lecture 2: Currents and their applications Lecture 3: Currents cont d.; Two words about parabolic implosion Lecture 5.5: Quasi-hyperbolicity
More informationQuasi-conformal maps and Beltrami equation
Chapter 7 Quasi-conformal maps and Beltrami equation 7. Linear distortion Assume that f(x + iy) =u(x + iy)+iv(x + iy) be a (real) linear map from C C that is orientation preserving. Let z = x + iy and
More informationHyperbolic geometry of Riemann surfaces
3 Hyperbolic geometry of Riemann surfaces By Theorem 1.8.8, all hyperbolic Riemann surfaces inherit the geometry of the hyperbolic plane. How this geometry interacts with the topology of a Riemann surface
More informationNotes on projective structures and Kleinian groups
Notes on projective structures and Kleinian groups Katsuhiko Matsuzaki and John A. Velling 1 Introduction Throughout this paper, C will denote the complex plane, Ĉ = C { } the number sphere, and D = {z
More informationLECTURE 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 informationCHAOTIC BEHAVIOR IN A TWO-DIMENSIONAL BUSINESS CYCLE MODEL
CHAOTIC BEHAVIOR IN A TWO-DIMENSIONAL BUSINESS CYCLE MODEL CRISTINA JANUÁRIO Department of Chemistry, Mathematics Unit, Instituto Superior de Engenharia de Lisboa, Rua Conselheiro Emídio Navarro, 1949-014
More information3 Fatou and Julia sets
3 Fatou and Julia sets The following properties follow immediately from our definitions at the end of the previous chapter: 1. F (f) is open (by definition); hence J(f) is closed and therefore compact
More informationMATH 311: COMPLEX ANALYSIS CONFORMAL MAPPINGS LECTURE
MATH 311: COMPLEX ANALYSIS CONFORMAL MAPPINGS LECTURE 1. Introduction Let D denote the unit disk and let D denote its boundary circle. Consider a piecewise continuous function on the boundary circle, {
More information2 hours THE UNIVERSITY OF MANCHESTER.?? January 2017??:????:??
hours MATH3051 THE UNIVERSITY OF MANCHESTER HYPERBOLIC GEOMETRY?? January 017??:????:?? Answer ALL FOUR questions in Section A (40 marks in all) and TWO of the THREE questions in Section B (30 marks each).
More informationMath 354 Transition graphs and subshifts November 26, 2014
Math 54 Transition graphs and subshifts November 6, 04. Transition graphs Let I be a closed interval in the real line. Suppose F : I I is function. Let I 0, I,..., I N be N closed subintervals in I with
More informationRandom Geodesics. Martin Bridgeman Boston College. n 1 denote the sphere at infinity. A Kleinian
Random Geodesics Martin Bridgeman Boston College 1 Background Let H n be n-dimensional hyperbolic space and let S n 1 denote the sphere at infinity. A Kleinian group is a discrete subgroup Γ of the group
More informationNewton s method and voronoi diagram
2015; 1(3): 129-134 ISSN Print: 2394-7500 ISSN Online: 2394-5869 Impact Factor: 3.4 IJAR 2015; 1(3): 129-134 www.allresearchjournal.com Received: 20-12-2014 Accepted: 22-01-2015 Anudeep Nain M. SC. 2nd
More informationKleinian groups Background
Kleinian groups Background Frederic Palesi Abstract We introduce basic notions about convex-cocompact actions on real rank one symmetric spaces. We focus mainly on the geometric interpretation as isometries
More informationTHE CHORDAL NORM OF DISCRETE MÖBIUS GROUPS IN SEVERAL DIMENSIONS
Annales Academiæ Scientiarum Fennicæ Mathematica Volumen 21, 1996, 271 287 THE CHORDAL NORM OF DISCRETE MÖBIUS GROUPS IN SEVERAL DIMENSIONS Chun Cao University of Michigan, Department of Mathematics Ann
More informationTopology Proceedings. COPYRIGHT c by Topology Proceedings. All rights reserved.
Topology Proceedings Web: http://topology.auburn.edu/tp/ Mail: Topology Proceedings Department of Mathematics & Statistics Auburn University, Alabama 36849, USA E-mail: topolog@auburn.edu ISSN: 0146-4124
More informationJunior Seminar: Hyperbolic Geometry Lecture Notes
Junior Seminar: Hyperbolic Geometry Lecture Notes Tim Campion January 20, 2010 1 Motivation Our first construction is very similar in spirit to an analogous one in Euclidean space. The group of isometries
More informationCounting geodesic arcs in a fixed conjugacy class on negatively curved surfaces with boundary
Counting geodesic arcs in a fixed conjugacy class on negatively curved surfaces with boundary Mark Pollicott Abstract We show how to derive an asymptotic estimates for the number of closed arcs γ on a
More informationPERIODIC POINTS ON THE BOUNDARIES OF ROTATION DOMAINS OF SOME RATIONAL FUNCTIONS
Imada, M. Osaka J. Math. 51 (2014), 215 224 PERIODIC POINTS ON THE BOUNDARIES OF ROTATION DOMAINS OF SOME RATIONAL FUNCTIONS MITSUHIKO IMADA (Received March 28, 2011, revised July 24, 2012) Abstract We
More information