What is a core and its entropy for a polynomial?

Transcription

1 e core-entropy := d dim(bi-acc(f )) What is a core and its entropy for a polynomial? Core(z 2 + c, c R) := K fc R Core(a polynomial f )=? Def.1. If exists, Core(f ) := Hubbard tree H, core-entropy:= h top (f, H) := log(lap-growth). Def.2. In all cases, Bi-acc(f ) := {angle-pairs in T 2 landing at a common point},

2 e core-entropy := d dim(bi-acc(f )) What is a core and its entropy for a polynomial? Core(z 2 + c, c R) := K fc R Core(a polynomial f )=? Def.1. If exists, Core(f ) := Hubbard tree H, core-entropy:= h top (f, H) := log(lap-growth). Def.2. In all cases, Bi-acc(f ) := {angle-pairs in T 2 landing at a common point},

3 e core-entropy := d dim(bi-acc(f )) What is a core and its entropy for a polynomial? Core(z 2 + c, c R) := K fc R Core(a polynomial f )=? Def.1. If exists, Core(f ) := Hubbard tree H, core-entropy:= h top (f, H) := log(lap-growth). Def.2. In all cases, Bi-acc(f ) := {angle-pairs in T 2 landing at a common point},

4 e core-entropy := d dim(bi-acc(f )) What is a core and its entropy for a polynomial? Core(z 2 + c, c R) := K fc R Core(a polynomial f )=? Def.1. If exists, Core(f ) := Hubbard tree H, core-entropy:= h top (f, H) := log(lap-growth). Def.2. In all cases, Bi-acc(f ) := {angle-pairs in T 2 landing at a common point},

5 e core-entropy := d dim(bi-acc(f )) What is a core and its entropy for a polynomial? Core(z 2 + c, c R) := K fc R Core(a polynomial f )=? Def.1. If exists, Core(f ) := Hubbard tree H, core-entropy:= h top (f, H) := log(lap-growth). Def.2. In all cases, Bi-acc(f ) := {angle-pairs in T 2 landing at a common point},

6 The Hubbard tree H c of f c : z z 2 + c, if exists, is the smallest topologically finite tree in K c containing the orbit of the critical point 0. It is automatically forward invariant.

7 The Hubbard tree H c of f c : z z 2 + c, if exists, is the smallest topologically finite tree in K c containing the orbit of the critical point 0. It is automatically forward invariant. H c exists for many "good" values of c. If postcritically finite, Markov partition and matrix M c and e htop(fc,hc) = λ leading (M c ).

8 Lemma : When H exists, e h top(f,h) dim(bi-acc(f )) = d The idea came from Milnor-Thurston s study on the real quadratic family f c (z) := z 2 + c c [ 2, 1 4 ], h(c) := h top(f c, [c, c 2 + c]) Theorem. c s(c) = exp(h(c)) is continuous (Milnor-Thurston) and weakly decreasing (Douady), 2 1. For the monotonicity, complex methods using external rays are more powerful...

9 Lemma : When H exists, e h top(f,h) dim(bi-acc(f )) = d The idea came from Milnor-Thurston s study on the real quadratic family f c (z) := z 2 + c c [ 2, 1 4 ], h(c) := h top(f c, [c, c 2 + c]) Theorem. c s(c) = exp(h(c)) is continuous (Milnor-Thurston) and weakly decreasing (Douady), 2 1. For the monotonicity, complex methods using external rays are more powerful...

10 (Complex methods) Douady s formula Y c := {t S 1, R c (t) lands on [c, c 2 +c]} is a closed invariant subset of q : t 2t, S 1 S 1, and airplane1.png h top (q, Y c ) = h(c). When 2 c, Y c, so h(c). A single function, various subsets...

11 W. Thurston s torus model, 2011 airplane1.png 2. Connect entropy to dimension : 1. A( single ) map ( ): s s F : 2, T 2 T 2. t t A c := {(t, 1 t) T 2, t Y c }, = bi-angles landing at a common point on R, is closed forward invariant set, with the same entropy. dim Ac Thurston 2 = e Core-entropy(fc) Tiozzo = 2 dim Yc. 3. Bi-acc(f c ) :=bi-angles landing at a common point on the Julia set, has A c as an attractor, and 2 dim Bi-acc(fc) = 2 dim Ac. Similarly 2 dim Proj S 1 (Bi-acc(f c)) Bruin-Schleicher = 2 dim Yc.

12 W. Thurston s torus model, 2011 airplane1.png 2. Connect entropy to dimension : 1. A( single ) map ( ): s s F : 2, T 2 T 2. t t A c := {(t, 1 t) T 2, t Y c }, = bi-angles landing at a common point on R, is closed forward invariant set, with the same entropy. dim Ac Thurston 2 = e Core-entropy(fc) Tiozzo = 2 dim Yc. 3. Bi-acc(f c ) :=bi-angles landing at a common point on the Julia set, has A c as an attractor, and 2 dim Bi-acc(fc) = 2 dim Ac. Similarly 2 dim Proj S 1 (Bi-acc(f c)) Bruin-Schleicher = 2 dim Yc.

13 W. Thurston s torus model, 2011 airplane1.png 2. Connect entropy to dimension : 1. A( single ) map ( ): s s F : 2, T 2 T 2. t t A c := {(t, 1 t) T 2, t Y c }, = bi-angles landing at a common point on R, is closed forward invariant set, with the same entropy. dim Ac Thurston 2 = e Core-entropy(fc) Tiozzo = 2 dim Yc. 3. Bi-acc(f c ) :=bi-angles landing at a common point on the Julia set, has A c as an attractor, and 2 dim Bi-acc(fc) = 2 dim Ac. Similarly 2 dim Proj S 1 (Bi-acc(f c)) Bruin-Schleicher = 2 dim Yc.

14 Combinatorial approach of W. Thurston Given d 2, consider the torus expanding covering ( ( ) F : T 2 T 2 s d s, mod 1 mod 1 t) d t W. Thurston : {B(m) non-closed, invariant subset m} core-entropy(polynomials). To define m, B(m) then Growth(m) := d dim(b(m)) A core T B(m) (if exists) is a closed invariant attractor. Results : core T = Growth(m)= ehtop(f,t ) P pcf = m P, Growth(m P )= e htop(p,hubbard-tree) m rational = core m rational = implement a matrix Γ m s.t. λ(γ m ) =Growth(m) Parameter space For any m, Growth(m) = exp h T 2(F, B(m)) in Bowen s sense

15 Combinatorial approach of W. Thurston Given d 2, consider the torus expanding covering ( ( ) F : T 2 T 2 s d s, mod 1 mod 1 t) d t W. Thurston : {B(m) non-closed, invariant subset m} core-entropy(polynomials). To define m, B(m) then Growth(m) := d dim(b(m)) A core T B(m) (if exists) is a closed invariant attractor. Results : core T = Growth(m)= ehtop(f,t ) P pcf = m P, Growth(m P )= e htop(p,hubbard-tree) m rational = core m rational = implement a matrix Γ m s.t. λ(γ m ) =Growth(m) Parameter space For any m, Growth(m) = exp h T 2(F, B(m)) in Bowen s sense

16 Combinatorial approach of W. Thurston Given d 2, consider the torus expanding covering ( ( ) F : T 2 T 2 s d s, mod 1 mod 1 t) d t W. Thurston : {B(m) non-closed, invariant subset m} core-entropy(polynomials). To define m, B(m) then Growth(m) := d dim(b(m)) A core T B(m) (if exists) is a closed invariant attractor. Results : core T = Growth(m)= ehtop(f,t ) P pcf = m P, Growth(m P )= e htop(p,hubbard-tree) m rational = core m rational = implement a matrix Γ m s.t. λ(γ m ) =Growth(m) Parameter space For any m, Growth(m) = exp h T 2(F, B(m)) in Bowen s sense

17 A primitive major (critical portrait) m of degree d = {disjoint hyperbolic leaves and ideal polygons in D} s.t. D m has d regions, each touches S 1 in a union of closed (non-point) intervals of total length 1/d. Tomasini : a closed form for number p d of combinatorial classes. d p d

18 A primitive major (critical portrait) m of degree d = {disjoint hyperbolic leaves and ideal polygons in D} s.t. D m has d regions, each touches S 1 in a union of closed (non-point) intervals of total length 1/d. Tomasini : a closed form for number p d of combinatorial classes. d p d

19 monic polynomial P? primitive major m Case 1. All critical points of P escape and escape with the same rate. Then P unique m by pulling back the external rays landing at the critical values and record only the rays landing at the critical points. Case 2. P is pcf. P finitely many m s. m is necessarily rational.

20 Primitive major m the binding set B(m) T 2 D m has d regions R i, with R i S 1 = a union of closed intervals J 1 J k of total length 1/d. Set T 2 G(m) := Ri (J 1 J k ) (J 1 J k ) (the light pink rectangles). The map F G(m) : G(m) T 2 is a degree d conformal repellor. Let K(m) be its non-escaping locus. Set B(m) := K(m) diagonal.

21 m G(m) and then B(m) whose dimension we want to measure In the quadratic and cubic cases : See deg2-deg3-toruswithlamination-growth

22 When a core T (closed inv. attractor subset of B(m)) exists Given d 2 and T C/Z 2 compact. Set ν(t, n) := # closed 1 d n Z2 -gridded tiles intersecting T ; log ν(t, n) if exists, D(T ) := lim n log d n. Theorem (Furstenberg or earlier?) For F : (x, y) (dx, dy), T closed, F (T ) T, D(T ) exists D(T ) log d = h top (F, T ) D(T ) = dim(t ) if = T =core dim(b(m)), d dim(b(m)) = e htop(f,t ).

23 When a core T (closed inv. attractor subset of B(m)) exists Given d 2 and T C/Z 2 compact. Set ν(t, n) := # closed 1 d n Z2 -gridded tiles intersecting T ; log ν(t, n) if exists, D(T ) := lim n log d n. Theorem (Furstenberg or earlier?) For F : (x, y) (dx, dy), T closed, F (T ) T, D(T ) exists D(T ) log d = h top (F, T ) D(T ) = dim(t ) if = T =core dim(b(m)), d dim(b(m)) = e htop(f,t ).

24 When a core T (closed inv. attractor subset of B(m)) exists Given d 2 and T C/Z 2 compact. Set ν(t, n) := # closed 1 d n Z2 -gridded tiles intersecting T ; log ν(t, n) if exists, D(T ) := lim n log d n. Theorem (Furstenberg or earlier?) For F : (x, y) (dx, dy), T closed, F (T ) T, D(T ) exists D(T ) log d = h top (F, T ) D(T ) = dim(t ) if = T =core dim(b(m)), d dim(b(m)) = e htop(f,t ).

25 Existence of growth rate For tiles intersecting T : {(n + m)-tiles} injects {(n-tile, m-tile)} S n F n F n T 2 S m S n+m So ν(t, n + m) ν(t, n) ν(t, m) and their growth rate exists.

26 The relation in the general setting, following W. Thurston post-cr-finite P rational m Hubbard tree H P e htop(p,h P) = λ(m(h P )) S 1 {angles landing on H P } d dim(angles) T 2 Bi-acc(P) d dim(bi-acc(p)) T 2 B(m) d dim B(m) comb. tree T (m) ehtop(f,t (m)) computable matrix Γ m λ(γ m ) T (m) = closure{leaves in B(m) separating the post-m angles} ; Γ m will be defined below. Theorem (proofs are being completed by Gao Y., see also W. Jung) T (m) is a core of B(m). All three lower right quantities are equal, and = top right three quantities.

27 The relation in the general setting, following W. Thurston post-cr-finite P rational m Hubbard tree H P e htop(p,h P) = λ(m(h P )) S 1 {angles landing on H P } d dim(angles) T 2 Bi-acc(P) d dim(bi-acc(p)) T 2 B(m) d dim B(m) comb. tree T (m) ehtop(f,t (m)) computable matrix Γ m λ(γ m ) T (m) = closure{leaves in B(m) separating the post-m angles} ; Γ m will be defined below. Theorem (proofs are being completed by Gao Y., see also W. Jung) T (m) is a core of B(m). All three lower right quantities are equal, and = top right three quantities.

28 The relation in the general setting, following W. Thurston post-cr-finite P rational m Hubbard tree H P e htop(p,h P) = λ(m(h P )) S 1 {angles landing on H P } d dim(angles) T 2 Bi-acc(P) d dim(bi-acc(p)) T 2 B(m) d dim B(m) comb. tree T (m) ehtop(f,t (m)) computable matrix Γ m λ(γ m ) T (m) = closure{leaves in B(m) separating the post-m angles} ; Γ m will be defined below. Theorem (proofs are being completed by Gao Y., see also W. Jung) T (m) is a core of B(m). All three lower right quantities are equal, and = top right three quantities.

29 W. Thurston s entropy matrix Γ {1/10,3/5} bypassing trees and dimensions (march 2011) Label the post-critical points by external angles. basis={pairs}= { { 1 5, 2 5 }, { 1 5, 3 5 }, { 1 5, 4 5 }, { 2 5, 3 5 }, { 2 5, 4 5 }, { 3 5, 4 5 }} Linear map Γ : { 1 5, 2 5 } { 2 5, 4 5 } { 1 5, 3 5 } { 2 5, 1 5 } { 1 5, 4 5 } { 1 5, 2 5 } + { 1 5, 3 5 } { 2 5, 3 5 } { 4 5, 1 5 } { 2 5, 4 5 } { 4 5, 1 5 } + { 1 5, 3 5 } { 3 5, 4 5 } { 1 5, 3 5 }

30 A similar algorithm works for any primitive major of any degree.

31 Q θ λ leading Γ( θ 2, θ+1 2 ), plot of W. Thurston Is this function continuous? Dyadic angles seem to be local maxima, true? It is l.s.c. (by W. Jung using results of Tiozzo about M). See also Bruin-Schleicher s arxiv paper.

32 Q θ λ leading Γ( θ 2, θ+1 2 ), plot of W. Thurston Is this function continuous? Dyadic angles seem to be local maxima, true? It is l.s.c. (by W. Jung using results of Tiozzo about M). See also Bruin-Schleicher s arxiv paper.

33 Q θ λ leading Γ( θ 2, θ+1 2 ), plot of W. Thurston Is this function continuous? Dyadic angles seem to be local maxima, true? It is l.s.c. (by W. Jung using results of Tiozzo about M). See also Bruin-Schleicher s arxiv paper.

34 two zooms at 1/6

35 Seeing Core-entropy from inside of M = {c C : f n c (0) }

36 The Mandelbrot set has a tree-like structure of veins, with the real segment as a particular vein. Theorem (Milnor-Thurston, Penrose,Li,Tiozzo,Jung, with a contribution of T.L.). Core-entropy is wake-monotone and continuous along the veins. W. Jung used this to prove the l.s.c. of θ λ leading (Γ mθ ). Entropy conjecture of Tiozzo.

37 Other models encoding core entropy Bartholdi-Dudko-Nekrashevych kneading automata, Tiozzo s extension of Milnor-Thurston s kneading determinant and uniform-expanding tree-maps, etc. See also works of Alsedà-Fagella, Branner-Hubbard, Dujardin-Favre, Milnor-Tesser, Penrose, Poirier, among others...

38 Other models encoding core entropy Bartholdi-Dudko-Nekrashevych kneading automata, Tiozzo s extension of Milnor-Thurston s kneading determinant and uniform-expanding tree-maps, etc. See also works of Alsedà-Fagella, Branner-Hubbard, Dujardin-Favre, Milnor-Tesser, Penrose, Poirier, among others...

39 Tiozzo s section theorem (2012) Set P c := {θ S 1 : the parameter θ-ray lands on [c, 1 4 [ }.

40 Tiozzo s section theorem (2012) Set P c := {θ S 1 : the parameter θ-ray lands on [c, 1 4 [ }.

41 Tiozzo s section theorem (2012) Set P c := {θ S 1 : the parameter θ-ray lands on [c, 1 4 [ }.

42 Tiozzo s section theorem (2012) Set P c := {θ S 1 : the parameter θ-ray lands on [c, 1 4 [ }. Then,

43 Tiozzo s section theorem (2012) Set P c := {θ S 1 : the parameter θ-ray lands on [c, 1 4 [ }. Then, e core-entropy(fc) = 2 dim Pc.

44 Theorem (W. Thurston, 2011). The space PM(d) embeds in the space of monic centered degree d (non-dynamical) polynomials as a spine for the set of polynomials with distinct roots, that is, the complement of the discriminant locus. The spine consists of polynomials whose critical values are all on the unit circle. Thus π 1 (PM(d)) is the d-strand braid group and all higher homotopy groups are trivial (i.e. PM(d) is a K(B d, 1) space). Proof. Take any degree d polynomial P with no multiple zeros, and look at log(p), thought of as a map from C \ roots to an infinite cylinder. For each critical point, draw the two separatrices going upward (i.e., this is the curve through each critical point of P that maps by P to a vertical half-line on the cylinder a ray in C pointed opposite the direction to the origin). Then make the finite lamination in a disk that joins the ending angles of these separatrices. This is a degree-d major set. Conversely, for each major set, there is a contractible family of polynomials whose separatrices end at the corresponding pairs of angles. To pick a canonical representative of each of these families : look at polynomials whose critical values are on the unit circle. This forms a spine for the complement of the discriminant locus for degree d polynomials.

45 Theorem (W. Thurston, 2011). The space PM(d) embeds in the space of monic centered degree d (non-dynamical) polynomials as a spine for the set of polynomials with distinct roots, that is, the complement of the discriminant locus. The spine consists of polynomials whose critical values are all on the unit circle. Thus π 1 (PM(d)) is the d-strand braid group and all higher homotopy groups are trivial (i.e. PM(d) is a K(B d, 1) space). Proof. Take any degree d polynomial P with no multiple zeros, and look at log(p), thought of as a map from C \ roots to an infinite cylinder. For each critical point, draw the two separatrices going upward (i.e., this is the curve through each critical point of P that maps by P to a vertical half-line on the cylinder a ray in C pointed opposite the direction to the origin). Then make the finite lamination in a disk that joins the ending angles of these separatrices. This is a degree-d major set. Conversely, for each major set, there is a contractible family of polynomials whose separatrices end at the corresponding pairs of angles. To pick a canonical representative of each of these families : look at polynomials whose critical values are on the unit circle. This forms a spine for the complement of the discriminant locus for degree d polynomials.

46 The space of cubic primitive majors

47 Growth of cubic primitive majors.

48 Bowen s definition of the relative entropy h T 2(F, B). Consider finite open covers A of T 2 and countable covers E of B. L A (E) := d n E :=separation time := d min{n, F n (E) a member of A} { dim L B := sup inf δ lim inf LA (E i ) δ = 0} A r 0 E,L A (E i )<r { dim B := inf δ lim Ei δ easy = 0} = dim L B, since r 0 inf E, E i <r E d min{n, F n (E) 1/d} d n E. D A (E) := e n E h(f, B) := sup A h(f, B) log d inf { h, lim inf DA (E i ) h = 0} r 0 E,D A (E i )<r = dim L B(= dim B) since D A (E) = L A (E) 1/ log d. If F (B) B, then h(f, B) h B (F, B)(:= sup on finite open covers A of B), since every A is an A. But h(f, T ) = h T (F, T ) if F (T ) T and T is compact, since a T -cover A becomes a T 2 -cover by adding T 2 T. So log ν(t, n) Furstenberg lim = dim T = h(f, T ) = h T (F, T ) Bowen = h top(f, T ). n n log d log d log d log d

49 Bowen s definition of the relative entropy h T 2(F, B). Consider finite open covers A of T 2 and countable covers E of B. L A (E) := d n E :=separation time := d min{n, F n (E) a member of A} { dim L B := sup inf δ lim inf LA (E i ) δ = 0} A r 0 E,L A (E i )<r { dim B := inf δ lim Ei δ easy = 0} = dim L B, since r 0 inf E, E i <r E d min{n, F n (E) 1/d} d n E. D A (E) := e n E h(f, B) := sup A h(f, B) log d inf { h, lim inf DA (E i ) h = 0} r 0 E,D A (E i )<r = dim L B(= dim B) since D A (E) = L A (E) 1/ log d. If F (B) B, then h(f, B) h B (F, B)(:= sup on finite open covers A of B), since every A is an A. But h(f, T ) = h T (F, T ) if F (T ) T and T is compact, since a T -cover A becomes a T 2 -cover by adding T 2 T. So log ν(t, n) Furstenberg lim = dim T = h(f, T ) = h T (F, T ) Bowen = h top(f, T ). n n log d log d log d log d

50 Bowen s definition of the relative entropy h T 2(F, B). Consider finite open covers A of T 2 and countable covers E of B. L A (E) := d n E :=separation time := d min{n, F n (E) a member of A} { dim L B := sup inf δ lim inf LA (E i ) δ = 0} A r 0 E,L A (E i )<r { dim B := inf δ lim Ei δ easy = 0} = dim L B, since r 0 inf E, E i <r E d min{n, F n (E) 1/d} d n E. D A (E) := e n E h(f, B) := sup A h(f, B) log d inf { h, lim inf DA (E i ) h = 0} r 0 E,D A (E i )<r = dim L B(= dim B) since D A (E) = L A (E) 1/ log d. If F (B) B, then h(f, B) h B (F, B)(:= sup on finite open covers A of B), since every A is an A. But h(f, T ) = h T (F, T ) if F (T ) T and T is compact, since a T -cover A becomes a T 2 -cover by adding T 2 T. So log ν(t, n) Furstenberg lim = dim T = h(f, T ) = h T (F, T ) Bowen = h top(f, T ). n n log d log d log d log d

51 Bowen s proof : F : T T compact, h top (F, T ) h T (F, T ) Let A : finite open cover of T, λ : lim inf DA (E i ) λ = 0. r 0 E cover of T,D A (E i )<r = countable cover E of T s.t. D A (E i ) λ < 1. m = finite open cover {D i } s.t. e λn i < 1 ; n i = n Di separation i=1 time of D i. = M := max n i, let function system {(D i, F n i )}, and its induced deeper pieces D j1 j 2 j s with separating time n j1 + + n js =: n D. Now A induces also puzzle pieces A ij indicating itineraries. n, {D, n D [n, n + M[} = a cover D n of T and each piece is contained in a A of level-n. A minimal cardinality sub cover among the level-n A s gives N(A n ) the cardinality. N(A n )e λn D D n e λn = D D n e λ(nd n) e λnd e Mλ all D e λnd <.

52 Thank you! Merci!

53 Relate to Milnor-Thurston s definition For n 0, θ(f n x +) := sgn(f n ) x + ( (f 0 = id). Generating function ) (κ A, κ B, κ C ) x +(t) := θ(f n x +) t n, in B, in C. (1 s A t)κ A = f n (x + ) A f n (x + ) A θ(f n x +) t n f n (x + ) A ( 1 sa t 1 s B t 1 s C t θ(f n+1 x +) t n+1 etc., adding the three : (κ A, κ B, κ C ) = = 1. n 0 n 1 ( (κa, κ B, κ C ) c + 1 knead. matrix N (t) = (Ã, B, C) = (κ ) A, κ B, κ C ) c 1 (κ A, κ B, κ C ) c + (κ A, κ B, κ C ) 2 c 2 Above (kernel vector)+cramer gives B C 1 s A t = Ã C 1 s B t = Ã B 1 s C t ) = kneading determinant D(t).

54 D old = D new in the unimodal case, with s A < 0, s B > 0 N (t) = (D B, D A ) := ( 1, +1) + ( n 1,f n (0 ± ) A ±2t n, n 1,f n (0 ± ) B ±2t n ) with + if f n has a local minimum and if local maximum. Then D B = (1 s B t)d new, D A = (1 s A t)d new. Since θ n = sgn(f n+1 ( ) = ) s interval sgn(f n ), ( ) sa t kernel vec. 1 2tD old = (D B, D A ) = (D s B t B, D A ) 1 = ( 1 + s B t + 1 s A t)d new = 2tD new. Or 1 2 R(t) = 1 2 (D A D B ) = 1 ( ) ( ) s DB, D A A = D 2 s old B = 1 + { ±ε(n)t n 1 if f with ε(n) = n (0 ± ) > 0 1 if f n (0 ± ) < 0. Clearly n 1 ε(n) determines whether f changes the type of extrema of f n (0).

55 Journées dynamiques holomorphes, Mai, Angers Confirmed speakers : R. Dujardin, N. Mihalache, P. Roesch, D.Thurston, G. Tiozzo, J. Tomasini