Entropy, Toulouse, 01-2014

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On W. Thurston’s core-entropy theory

Tan Lei 谭蕾 Université d’Angers

(most figures/codes are produced by William Thurston and Giulio Tiozzo)

Toulouse 图卢兹, January, 2014

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Givend ≥2, consider the torus expanding covering F:T²→T²,

s t

mod 17→

d·s d·t

mod 1

W. Thurston :{B(m) non-closed, invariant subset|m}

−→ core-entropy(polynomials).

To define

• m,B(m) then

• Growth(m):= d^dim(B^(m))

• AcoreT ⊂B(m) (if exists) is a ’closed’ invariant attractor.

Results :

• ∃ coreT =⇒Growth(m)=e^h^top^(F^,T)

• P pcf =⇒ ∃m_P,Growth(m_P)=e^h^top(P,Hubbard-tree)

• mrational =⇒ ∃ core

• mrational =⇒implement a matrix Γm s.t. λ(Γm) =Growth(m)

• Parameter space

• For anym, Growth(m) =exph

T²(F,B(m))in Bowen’s sense

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s t

mod 17→

d·s d·t

mod 1

To define

• m,B(m) then

Results :

• Parameter space

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s t

mod 17→

d·s d·t

mod 1

To define

• m,B(m) then

Results :

• Parameter space

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A primitive majorm of degreed =

{disjoint hyperbolic leaves and ideal polygons inD} s.t.D rm has d regions, each touchesS¹ in a union of closed (non-point)

intervals of total length 1/d.

Tomasini: a closed form for numberp˜_d of combinatorial classes.

d 2 3 4 5 6 7 8 9 10 11 12

˜

p_d 1 2 4 9 27 94 364 1529 6689 30230 140114 · · ·

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A primitive majorm of degreed =

{disjoint hyperbolic leaves and ideal polygons inD} s.t.D rm has d regions, each touchesS¹ in a union of closed (non-point)

intervals of total length 1/d.

Tomasini: a closed form for numberp˜_d of combinatorial classes.

d 2 3 4 5 6 7 8 9 10 11 12

˜

p_d 1 2 4 9 27 94 364 1529 6689 30230 140114 · · ·

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monic polynomial P −→ primitive major m

Case 1. All critical points ofP escape and escape with the same rate. ThenP ←→a uniquem 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 a postcritically finite polynomial, or more generally a connected locally connected filled Julia set. Then

P −→finitely manym’s by choosing one preferred ray and its co-rays per Fatou-cr-component and per Julia cr. point (so that the preferred rays keep the corresponding critical relations). In the postcritically finite casemis necessarily rational.

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Primitive major m −→ the binding set B (m) ⊂ T

²

D rm has d regions R_i, withR_i∩S¹ =a union of closed intervals J₁∪ · · · ∪J_k of total length 1/d. Set

T² ⊃G(m) :=∪_R_i(J₁∪ · · · ∪J_k)×(J₁∪ · · · ∪J_k) (the light pink rectangles). The mapF|_G_(m):G(m)→T² is a degree d

’conformal repellor’. LetK(m) be its non-escaping locus. Set B(m) :=K(m)rdiagonal.

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m −→ G (m) and then B (m) whose dimension we want to measure

In the quadratic and cubic cases :

See deg2-deg3-TorusWithLamination-Growth

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When a core T (closed inv. attractor subset of B (m)) exists

Given d ≥ 2 and T ⊂ C/Z² compact.

Set

•ν(T,n) := #closed 1

dⁿZ²-gridded tiles intersectingT;

•if exists, D(T):= lim

n→∞

logν(T,n) logdⁿ . Theorem(Furstenberg or earlier ?)

ForF : (x,y)7→(dx,dy),T closed,F(T)⊂T,

•D(T) exists

•D(T)·logd =htop(F,T)

•D(T)=dim(T) =^if

T=core dim(B(m)), d^dim(B^(m))=e^h^top^(F^,T) .

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When a core T (closed inv. attractor subset of B (m)) exists

Set

n→∞

•D(T) exists

•D(T)=dim(T) =^if

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When a core T (closed inv. attractor subset of B (m)) exists

Set

n→∞

•D(T) exists

•D(T)=dim(T) =^if

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Existence of growth rate

For tiles intersectingT :

{(n+m)-tiles}^injects,→ {(n-tile,m-tile)}

T²

%_F^◦n ∪

S_n S_m

∪ %_F^◦n S_n+m

Soν(T,n+m)≤ν(T,n)·ν(T,m) and their growth rate exists.

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What is a core and its entropy for a polynomial ?

Core(z²+c, c ∈R) := K_f_c∩R Core(a polynomialf)= ?

If exists, Core(f) :=Hubbard tree H, core-entropy:=h_top(f,H).

Or, Bi-acc(f) :=

{angle-pairs inT² landing at a common point}, ecore-entropy :=ddim(Bi-acc(f))

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What is a core and its entropy for a polynomial ?

Or, Bi-acc(f) :=

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What is a core and its entropy for a polynomial ?

If exists,Core(f) :=Hubbard tree H, core-entropy:=h_top(f,H).

Or, Bi-acc(f) :=

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What is a core and its entropy for a polynomial ?

Or,Bi-acc(f) :=

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What is a core and its entropy for a polynomial ?

Or, Bi-acc(f) :=

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TheHubbard treeH_c of f_c :z 7→z²+c, if exists, is the smallest topologically finite tree inK_c containing the orbit of the critical point 0. It is automatically forward invariant.

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TheHubbard treeH_c of f_c :z 7→z²+c, if exists, is the smallest topologically finite tree inK_c containing the orbit of the critical point 0. It is automatically forward invariant.

H_c exists for many "good" values ofc.

If postcritically finite,∃ Markov partition and matrixM_c and e^h^top^(f^c^,H^c⁾=λ_leading(M_c).

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When H exists, e

^h^top^(f^,H)

= d

dim(Bi-acc(f))

The story began with Milnor-Thurston’s study on the real quadratic family (’77)

f_c(z) :=z²+c c ∈[−2,1

4], h(c) :=h_top(f_c,[c,c²+c])

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When H exists, e

^h^top^(f^,H)

= d

dim(Bi-acc(f))

The story began with Milnor-Thurston’s study on the real quadratic family (’77)

f_c(z) :=z²+c c ∈[−2,1

4], h(c) :=h_top(f_c,[c,c²+c]) (Complex methods)Douady’s formula

Yc :={t∈S¹, Rc(t)lands on[c,c²+c]}

is a closed invariant subset of q:t 7→2t, S¹→S¹, and

htop(q,Yc) =h(c).

Yc &, soh(c)&.

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W. Thurston’s bi-circle (torus) point of view, 2011

1. The setAc :={(t,1−t)∈T²,t ∈Yc},

= bi-angles landing at a common point on R, is a closed forward invariant set of F : s

t

7→2 s

t

,T²→T²with thesame entropy.

2. One can then connectentropy to dimension: 2^dimA^c ^Thurston= eCore-entropy(fc) Tiozzo

= 2^dim^Y^c .

3. Bi-acc(f_c):=bi-angles landing at a common point on the Julia set, has Ac as a core. Consequently

2dim Bi-acc(fc) =2^dim^A^c. Similarly 2^dimProj^S¹^(Bi-acc(f^c⁾⁾Bruin-Schleicher

= 2^dim^Y^c.

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W. Thurston’s bi-circle (torus) point of view, 2011

t

7→2 s

t

= 2^dim^Y^c .

= 2^dim^Y^c.

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W. Thurston’s bi-circle (torus) point of view, 2011

t

7→2 s

t

= 2^dim^Y^c .

= 2^dim^Y^c.

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The relation in the general setting, following W. Thurston

post-cr-finiteP −→

Hubbard treeH_P −→e^h^top^(P,H^P⁾ =λ(M(H_P)) S¹⊃ {angles landing onH_P} →ddim(angles)

T² ⊃Bi-acc(P)−→ddim(Bi-acc(P))

l − − − − − − − − − − − − − − − − −

rationalm −→

T² ⊃B(m) −→ d^dim^B^(m) comb. treeT(m) −→ e^h^top^(F^,T(m)) computable matrixΓm −→ λ(Γm) T(m) = closure{leaves inB(m) separating the post-m angles};Γm

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.

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The relation in the general setting, following W. Thurston

l − − − − − − − − − − − − − − − − −

rationalm −→

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The relation in the general setting, following W. Thurston

l − − − − − − − − − − − − − − − − −

rationalm −→

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W. Thurston’s entropy matrixΓ_{1/10,3/5} bypassing trees and dimensions (march 2011)

Label the post-critical points by external angles.

basis={pairs}=

{¹₅,²₅},{¹₅,³₅},{¹₅,⁴₅},{²₅,³₅},{²₅,⁴₅},{³₅,⁴₅} Linear mapΓ :

{¹₅,²₅} → {²₅,⁴₅} {¹₅,³₅} → {²₅,¹₅} {¹₅,⁴₅} → {¹₅,²₅}+{¹₅,³₅} {²₅,³₅} → {⁴₅,¹₅} {²₅,⁴₅} → {⁴₅,¹₅}+{¹₅,³₅} {³₅,⁴₅} → {¹₅,³₅}

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Start from any rational angleθ6=0, the primitive major

m=leaf(θ/2, θ+1/2)separate the circle into two halves, each half is a closed segment in the circle (so the cutting points belong to both halves).

1. Form the set of all possible unordered pairs{2ⁿθ,2^lθ}with l,n ≥0 and the two angles are unequal. Consider them as a basis of an abstract linear space.

2. Compute the linear mapΓm and its leading eigenvalue :

{2ⁿθ,2^lθ} 7→

( {2ⁿ⁺¹θ,2^l+1θ} if 2ⁿθ,2^lθ are in the same half-circle {2ⁿ⁺¹θ, θ}+{θ,2^l+1θ} otherwise

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

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Q3θ7→λ_leadingΓ(^θ₂,^θ+1₂ ), 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 aboutM). See also Bruin-Schleicher’s arxiv paper.

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Is this function continuous ? Dyadic angles seem to be local maxima, true ?It is l.s.c. (by W. Jung using results of Tiozzo aboutM). See also Bruin-Schleicher’s arxiv paper.

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Is this function continuous ? Dyadic angles seem to be local maxima, true ? It is l.s.c. (by W. Jung using results of Tiozzo aboutM). See also Bruin-Schleicher’s arxiv paper.

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two zooms at 1/6

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Seeing Core-entropy from

M={c ∈C : f_c^◦n(0)9∞}

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Tiozzo’s section theorem (2012)

SetP_c :={θ∈S¹ : the parameterθ-ray lands on[c,1 4[}.

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Tiozzo’s section theorem (2012)

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Tiozzo’s section theorem (2012)

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Tiozzo’s section theorem (2012)

Then,

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Tiozzo’s section theorem (2012)

Then, ecore-entropy(fc)=2^dim^P^c.

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The Mandelbrot set has a tree-like structure ofveins, with the real segment as a particular vein.

Theorem(Milnor-Thurston, Penrose,Li,Tiozzo,Jung, with a contribution of T.L.). Core-entropy iswake-monotoneand continuous along the veins.

W. Jung used this to prove the l.s.c. ofθ7→λ_leading(Γm_θ). Entropy conjecture of Tiozzo.

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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...

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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...

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Theorem(W. Thurston, 2011). The spacePM(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 π1(PM(d))is thed-strand braid group and all higher homotopy groups are trivial (i.e.PM(d) is aK(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 fromC\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 inCpointed 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.

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Theorem(W. Thurston, 2011). The spacePM(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 π1(PM(d))is thed-strand braid group and all higher homotopy groups are trivial (i.e.PM(d) is aK(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 fromC\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 inCpointed 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.

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The space of cubic primitive majors

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Growth of cubic primitive majors.

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Bowen’s definition of the relative entropyh

T²(F,B).

Consider finite open coversAofT²and countable coversE ofB. LA(E) := d⁻ⁿ^E:=separation time

:=d^−min{n,^Fⁿ^(E)6⊂a member ofA}

dimLB := sup

A

infn δ| lim

r→0 inf

E,LA(E_i)<r

XLA(Ei)^δ=0o

dimB := infn δ| lim

r→0 inf

E,|Ei|<r

X|Ei|^δ=0o_easy

= dimLB, since

|E| d^−min{n,|Fⁿ^(E^)|≥1/d}d⁻ⁿ^E. DA(E) := e⁻ⁿ^E

h(F,B) := sup

A

infn h,lim

r→0 inf

E,DA(E_i)<r

XD_A(Ei)^h=0o h(F,B)

logd = dimLB(=dimB) since DA(E) =LA(E)^1/^log^d.

IfF(B)⊂B, thenh(F,B)≤hB(F,B)(:=sup on finite open coversA⁰ ofB), sinceeveryAis anA⁰. Buth(F,T) =hT(F,T)ifF(T)⊂T andT is compact, since aT-coverA⁰ becomes aT²-cover by addingT²rT. So

n→∞lim

logν(T,n) nlogd

Furstenberg

= dimT = h(F,T)

logd = hT(F,T) logd

Bowen

= htop(F,T) logd .

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T²(F,B).

dimLB := sup

A

infn δ| lim

r→0 inf

E,LA(E_i)<r

XLA(Ei)^δ=0o

r→0 inf

E,|Ei|<r

X|Ei|^δ=0o_easy

= dimLB, since

h(F,B) := sup

A

infn h,lim

r→0 inf

E,DA(E_i)<r

IfF(B)⊂B, thenh(F,B)≤hB(F,B)(:=sup on finite open coversA⁰ ofB), sinceeveryAis anA⁰.Buth(F,T) =hT(F,T)ifF(T)⊂T andT is compact, since aT-coverA⁰ becomes aT²-cover by addingT²rT. So

n→∞lim

logν(T,n) nlogd

Furstenberg

= dimT = h(F,T)

logd = hT(F,T) logd

Bowen

= htop(F,T) logd .

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T²(F,B).

dimLB := sup

A

infn δ| lim

r→0 inf

E,LA(E_i)<r

XLA(Ei)^δ=0o

r→0 inf

E,|Ei|<r

X|Ei|^δ=0o_easy

= dimLB, since

h(F,B) := sup

A

infn h,lim

r→0 inf

E,DA(E_i)<r

IfF(B)⊂B, thenh(F,B)≤hB(F,B)(:=sup on finite open coversA⁰ ofB), sinceeveryAis anA⁰. Buth(F,T) =hT(F,T)ifF(T)⊂T andT is compact, since aT-coverA⁰ becomes aT²-cover by addingT²rT. So

n→∞lim

logν(T,n) nlogd

Furstenberg

= dimT = h(F,T)

logd = hT(F,T) logd

Bowen

= htop(F,T) logd .

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Bowen’s proof :F :T →T compact, h_top(F,T)≤h_T(F,T) LetA: finite open cover of T,

λ: lim

r→0 inf

Ecover ofT,DA(Ei)<r

XDA(Ei)^λ=0.

=⇒ ∃countable coverE of T s.t. P

DA(E_i)^λ <1.

=⇒finite open cover {D_i} s.t.

m

X

i=1

e^−λnⁱ <1 ; n_i =n_D_i separation time ofD_i.

=⇒M :=maxn_i, let function system{(D_i,Fⁿⁱ)}, and its induced deeper piecesD_j₁_j₂_···j_s with separating time≥n_j₁+· · ·+n_j_s =:n_D_∗. NowA induces also puzzle piecesAij··· indicating itineraries.

∀n,{D∗,n_D_∗ ∈[n,n+M[}= a coverD_n ofT and each piece is contained in aA∗ of level-n. A minimal cardinality sub cover among the level-n A∗’s gives N(A_n)the cardinality.

N(An)e^−λn≤ X

D∗∈Dn

e^−λn= X

D∗∈Dn

e^λ(n^D∗⁻ⁿ⁾e^−λn^D∗ ≤e^MλX

allD∗

e^−λn^D∗<∞.

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Thank you ! Merci !