Entropy, Orleans, 03-2013

Core entropy of real and complex polynomials 实与 复多项式 的核 拓扑 熵

Tan Lei 谭蕾 Université d’Angers

(most figures are provided by Giulio Tiozzo)

Orléans奥尔良, March 12, 2013

Fix a degreed≥2. For everyn, partition the unit square into d²ⁿ small squares of size 1

dⁿ. For a closed subset A, denote by ν(A,n)the number of standard level- n squares intersecting A and, whenever the limit exist,

h(A):=limn→∞

logν(A,n)

n ,

d(A):= lim

n→∞

logν(A,n) logsize(level-n square)¹

= lim

n→∞

logν(A,n) n logd

trivially

= h(A) logd Theorem(adapting Furstenberg’s similar result on S¹)

ForF :T² →T²,(x,y)7→(dx,dy),F(A)⊂A,Aclosed,

•both limits exist

•h(A)=htop(F|_A) (topological entropy, defined for continuous f :X →X on compactX),

•d(A)=H.dim(A),d(A) = h(A) logd.

Fix a degreed≥2. For everyn, partition the unit square into d²ⁿ small squares of size 1

dⁿ. For a closed subset A, denote by ν(A,n)the number of standard level- n squares intersecting A and, whenever the limit exist,

h(A):=limn→∞

logν(A,n)

n ,

d(A):= lim

n→∞

logν(A,n) logsize(level-n square)¹

= lim

n→∞

logν(A,n) n logd

trivially

= h(A) logd Theorem(adapting Furstenberg’s similar result on S¹)

ForF :T² →T²,(x,y)7→(dx,dy),F(A)⊂A,Aclosed,

•both limits exist

•h(A)=htop(F|_A) (topological entropy, defined for continuous f :X →X on compactX),

•d(A)=H.dim(A),d(A) = h(A) logd.

Fix a degreed≥2. For everyn, partition the unit square into d²ⁿ small squares of size 1

dⁿ. For a closed subset A, denote by ν(A,n)the number of standard level- n squares intersecting A and, whenever the limit exist,

h(A):=limn→∞

logν(A,n)

n ,

d(A):= lim

n→∞

logν(A,n) logsize(level-n square)¹

= lim

n→∞

logν(A,n) n logd

trivially

= h(A) logd Theorem(adapting Furstenberg’s similar result on S¹)

ForF :T² →T²,(x,y)7→(dx,dy),F(A)⊂A,Aclosed,

•both limits exist

•h(A)=htop(F|_A) (topological entropy, defined for continuous f :X →X on compactX),

•d(A)=H.dim(A),d(A) = h(A) logd.

Fix a degreed≥2. For everyn, partition the unit square into d²ⁿ small squares of size 1

dⁿ. For a closed subset A, denote by ν(A,n)the number of standard level- n squares intersecting A and, whenever the limit exist,

h(A):=limn→∞

logν(A,n)

n ,

d(A):= lim

n→∞

logν(A,n) logsize(level-n square)¹

= lim

n→∞

logν(A,n) n logd

trivially

= h(A) logd

Theorem(adapting Furstenberg’s similar result on S¹) ForF :T² →T²,(x,y)7→(dx,dy),F(A)⊂A,Aclosed,

•both limits exist

•h(A)=htop(F|_A) (topological entropy, defined for continuous f :X →X on compactX),

•d(A)=H.dim(A),d(A) = h(A) logd.

Fix a degreed≥2. For everyn, partition the unit square into d²ⁿ small squares of size 1

dⁿ. For a closed subset A, denote by ν(A,n)the number of standard level- n squares intersecting A and, whenever the limit exist,

h(A):=limn→∞

logν(A,n)

n ,

d(A):= lim

n→∞

logν(A,n) logsize(level-n square)¹

= lim

n→∞

logν(A,n) n logd

trivially

= h(A) logd Theorem(adapting Furstenberg’s similar result on S¹)

ForF :T² →T²,(x,y)7→(dx,dy),F(A)⊂A,Aclosed,

•both limits exist

•h(A)=htop(F|_A) (topological entropy, defined for continuous f :X →X on compactX),

•d(A)=H.dim(A),d(A) = h(A) logd.

Existence of the limit

{Sn+m},→ {(Sn,Sm)}

T²

%_F◦n ∪

Sn Sm

∪ %_F◦n

Sn+m

Soν(A,n+m)≤ν(A,n)·ν(A,m).This sub-multiplicativity implies thath(A) andd(A) exist.

Corollary. ForF :T² →T², x

y

7→d x

y

,F(A)⊂A,Aclosed, all sets· · · ⊃F⁻ⁿ(A)⊃ · · · ⊃F⁻¹(A)⊃A⊃F(A)⊃F²(A)⊃ · · · have the same dimension and entropy, and their union

B(A) =S

nF⁻ⁿ(A) has the same dimension.

We will see that(F,A),(F,B(A))for various Aparametrizes combinatorially all degreed polynomials, with h(A) as the core entropy.

Existence of the limit

{Sn+m},→ {(Sn,Sm)}

T²

%_F◦n ∪

Sn Sm

∪ %_F◦n

Sn+m

Soν(A,n+m)≤ν(A,n)·ν(A,m).This sub-multiplicativity implies thath(A) andd(A) exist.

Corollary. ForF :T² →T², x

y

7→d x

y

,F(A)⊂A,Aclosed, all sets· · · ⊃F⁻ⁿ(A)⊃ · · · ⊃F⁻¹(A)⊃A⊃F(A)⊃F²(A)⊃ · · · have the same dimension and entropy, and their union

B(A) =S

nF⁻ⁿ(A) has the same dimension.

We will see that(F,A),(F,B(A))for various Aparametrizes combinatorially all degreed polynomials, with h(A) as the core entropy.

Core-entropy of a real polynomial f = entropy(f | R )

Letf :I →I continuous, by Misiurewicz-Szlenk, htop(f,R) := lim

n→∞

log{#laps of fⁿ} n

Core-entropy of a real polynomial f = entropy(f | R )

Letf :I →I continuous, by Misiurewicz-Szlenk, htop(f,R) := lim

n→∞

log{#laps of fⁿ} n

Core-entropy of a real polynomial f = entropy(f | R )

Letf :I →I continuous, by Misiurewicz-Szlenk, htop(f,R) := lim

n→∞

log{#laps of fⁿ} n

Core-entropy of a real polynomial f = entropy(f | R )

Letf :I →I continuous, by Misiurewicz-Szlenk, htop(f,R) := lim

n→∞

log{#laps of fⁿ} n

Core-entropy of a real polynomial f = entropy(f | R )

Letf :I →I continuous, by Misiurewicz-Szlenk, htop(f,R) := lim

n→∞

log{#laps of fⁿ} n

Core-entropy of a real polynomial f = entropy(f | R )

Letf :I →I continuous, by Misiurewicz-Szlenk, htop(f,R) := lim

n→∞

log{#laps of fⁿ} n

Core-entropy of a real polynomial f = entropy(f | R )

Letf :I →I continuous, by Misiurewicz-Szlenk, htop(f,R) := lim

n→∞

log{#laps of fⁿ} n

Core-entropy of a real polynomial f = entropy(f | R )

Letf :I →I continuous, by Misiurewicz-Szlenk, htop(f,R) := lim

n→∞

log{#laps of fⁿ} n

Parameter variation, for the real quadratic family

fc(z) :=z²+c c ∈[−2,1/4]

how to computehtop(fc,R) =h(c)? How does it change with the parameterc? By Milnor-Thurston (77),

I h(c) iscontinuousandmonotone decreasing, log 2&0

I Thethe kneading seriesDc(t) =P

k≥0±t^k

+iffc^k has a local minimum at 0, and−local maximum admits 1/e^h(c) as a zero with least modulus.

I The map x 7→K(x) =P

k±λ^−k

+(−) iff^k|_x %(&) , is continuous iff λ=e^h(c) and semi-conjugatesf to the tent map of slope e^h(c).

Parameter variation, for the real quadratic family

fc(z) :=z²+c c ∈[−2,1/4]

how to computehtop(fc,R) =h(c)? How does it change with the parameterc?

By Milnor-Thurston (77),

I h(c) iscontinuousandmonotone decreasing, log 2&0

I Thethe kneading seriesDc(t) =P

k≥0±t^k

+iffc^k has a local minimum at 0, and−local maximum admits 1/e^h(c) as a zero with least modulus.

I The map x 7→K(x) =P

k±λ^−k

+(−) iff^k|_x %(&) , is continuous iff λ=e^h(c) and semi-conjugatesf to the tent map of slope e^h(c).

Parameter variation, for the real quadratic family

fc(z) :=z²+c c ∈[−2,1/4]

how to computehtop(fc,R) =h(c)? How does it change with the parameterc? By Milnor-Thurston (77),

I h(c) iscontinuousandmonotone decreasing, log 2&0

I Thethe kneading seriesDc(t) =P

k≥0±t^k

+iffc^k has a local minimum at 0, and−local maximum admits 1/e^h(c) as a zero with least modulus.

I The map x 7→K(x) =P

k±λ^−k

+(−) iff^k|_x %(&) , is continuous iff λ=e^h(c) and semi-conjugatesf to the tent map of slope e^h(c).

Parameter variation, for the real quadratic family

fc(z) :=z²+c c ∈[−2,1/4]

how to computehtop(fc,R) =h(c)? How does it change with the parameterc? By Milnor-Thurston (77),

I h(c) iscontinuousandmonotone decreasing, log 2&0

I Thethe kneading seriesDc(t) =P

k≥0±t^k

+ iffc^k has a local minimum at 0, and−local maximum admits 1/e^h(c) as a zero with least modulus.

I The map x 7→K(x) =P

k±λ^−k

+(−) iff^k|_x %(&) , is continuous iff λ=e^h(c) and semi-conjugatesf to the tent map of slope e^h(c).

Complex view of real z

+ c , Douady’s circle model

Eachfc has a Julia setKc :={z ∈C|fc^◦n(z)9∞}.

Complex view of real z

+ c , Douady’s circle model

Eachfc has a Julia setKc :={z ∈C|fc^◦n(z)9∞}. Forc ∈[−2,¹₄], the setC rˆ Kc is simply-

connected, there is a conformal mapΦc : C r Dˆ → C rˆ Kc semi-conjugating z² to fc. The external rays{Rc(t),t ∈ S¹} co- ming from radial rays outsideD, foliate the outside of Kc, and are invariant by f : fc(Rc(t)) = Rc(2t). This translates the core entropy to an entropy of thesymbolic dynamics:

Complex view of real z

+ c , Douady’s circle model

Eachfc has a Julia setKc :={z ∈C|fc^◦n(z)9∞}. Forc ∈[−2,¹₄], the setC rˆ Kc is simply-

connected, there is a conformal mapΦc : C r Dˆ → C rˆ Kc semi-conjugating z² to fc. The external rays{Rc(t),t ∈ S¹} co- ming from radial rays outsideD, foliate the outside of Kc, and are invariant by f : fc(Rc(t)) = Rc(2t). This translates the core entropy to an entropy of thesymbolic dynamics:

Douady’s formula (’93):Yc :={t ∈S¹ : Rc(t) lands on R} is a closed invariant subset of theangle doubling map

q :t 7→2t, S¹ →S¹, and

htop(q,Yc) =htop(fc,R) =:Core-entropy(fc).

Thurston’s bi-circle (torus) model, ’2001

The set Ac := {(t,1−t) ∈ T²,t ∈ Yc}, consisting ofbi-angles landing at a common point onR, is a closed forward invariant set of F :

s t

7→ 2 s

t

, T² → T² with the same entropy.

One can then connectentropy to dimensionvia Furstenberg : H.dim(Ac)^Thurston= Core-entropy(fc)

log 2

Tiozzo

= H.dim(Yc) . Consequently, the set B(Ac) :=S

nF⁻ⁿ(Ac), consisting of

bi-angles landing at a common point on the Julia set, has the same dimension asAc.

Thurston’s bi-circle (torus) model, ’2001

The set Ac := {(t,1−t) ∈ T²,t ∈ Yc}, consisting ofbi-angles landing at a common point onR, is a closed forward invariant set of F :

s t

7→ 2 s

t

, T² → T² with the same entropy.

One can then connectentropy to dimensionvia Furstenberg : H.dim(Ac)^Thurston= Core-entropy(fc)

log 2

Tiozzo

= H.dim(Yc) .

Consequently, the set B(Ac) :=S

nF⁻ⁿ(Ac), consisting of

bi-angles landing at a common point on the Julia set, has the same dimension asAc.

Thurston’s bi-circle (torus) model, ’2001

The set Ac := {(t,1−t) ∈ T²,t ∈ Yc}, consisting ofbi-angles landing at a common point onR, is a closed forward invariant set of F :

s t

7→ 2 s

t

, T² → T² with the same entropy.

One can then connectentropy to dimensionvia Furstenberg : H.dim(Ac)^Thurston= Core-entropy(fc)

log 2

Tiozzo

= H.dim(Yc) . Consequently, the set B(Ac) :=S

nF⁻ⁿ(Ac), consisting of

bi-angles landing at a common point on the Julia set, has the same dimension asAc.

Parameter view of real z

+ c

TheMandelbrot setMis defined by

M={c ∈C : fc^◦n(0)9∞}

External rays

SinceCˆ \ M is simply-connected (by Douady-Hubbard and Sibony), there is a conformal map :

ΦM: ˆC\D→Cˆ \ M

parametrizing the outside and the boundary ofMby external angles.

Almost all (and conjecturally all) rays land at some point on the boundary ofM.

Douady’s monotonicity argument (’93)

The following chain of associations are trivially monotone : [−2,1

4]3c external angle

−→ θ∈[0,1

2]→forbidden region]θ,1−θ[

−→ Xθ :={t∈S¹ |2ⁿt∈/ ]θ,1−θ[}

−→ htop(q,Xθ)^q(Yc=^)=Xθ

less trivial core-entropy(fc).

Tiozzo’s section theorem (’2012)

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

Tiozzo’s section theorem (’2012)

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

Tiozzo’s section theorem (’2012)

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

Tiozzo’s section theorem (’2012)

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

Then,

Tiozzo’s section theorem (’2012)

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

Then, core-entropy(fc)

log 2 =H.dimPc.

Tiozzo’s section theorem (’2012)

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

Then, core-entropy(fc)

log 2 =H.dimPc.

The proof uses techniques coming fromα-continued fractions.

How to define the core entropy for a non-real polynomial P ?

One can define it to beH.dim(B(P))·log degree, for

B(P) :=the set of bi-angles landing at a common point, this is well defined for anyP, or

htop(P,H), if there exists aforward invariant tree H playing the role of the real line ...

How to define the core entropy for a non-real polynomial P ?

One can define it to beH.dim(B(P))·log degree, for

B(P) :=the set of bi-angles landing at a common point, this is well defined for anyP, or

htop(P,H), if there exists aforward invariant tree H playing the role of the real line ...

How to define the core entropy for a non-real polynomial P ?

One can define it to beH.dim(B(P))·log degree, for

B(P) :=the set of bi-angles landing at a common point, this is well defined for anyP, or

htop(P,H), if there exists aforward invariant tree H playing the role of the real line ...

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

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

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

For many "good" values ofc, the Hubbard tree, and many larger forward invariant trees, exist, and all give the same entropy (see Tiozzo’s preprint). For even better values ofc (such that 0 has a finite orbit), the tree admits aMarkov partition and matrix and

Core-entropy(fc) =logλleading(Markov matrix).

Other models

There are other means to encode the core entropy, such as Bartholdi-Dudko-Nekrashevych automata model, Tiozzo’s

parameter section model, Milnor-Thurston’s kneading determinant and uniform-expander model, etc. Relating them all and studying parameter variations is part of the current research. See work of Alsedà-Fagella, Bartholdi-Dudko-Nekrashevych, Branner-Hubbard, Bruin-Schleicher, Dujardin-Favre, Gao Y.高延, Li T.李涛, Milnor-Tesser, Penrose, Poirier, Thurston, Tiozzo, among others...

Thurston’s parametrization of degreed polynomials by primitive majors (critical portrait) (partially proved by Gao and T.)

P → Critical portraitm →Forbidden regions

−→ T² ⊃K(m) :=the non escaping set

−→ B(m) :=K(m)rdiagonal

−→

For goodm B(m) =[

n

F⁻ⁿ(A(m)),A(m) closed and invariant

computable

→

For rationalm matrix Γm whose λleading Thm=

Gao exp(entropy).

From a primitive major m to its binding set B ( m )

A primitive majorm of degreed is a finite lamination in Dso that each region ofD rmtouches S¹ in a union of one or more closed intervalsJ1∪ · · · ∪Jk of total length 1/d. The product

(J1∪ · · · ∪Jk)×(J1∪ · · · ∪Jk) is a union of light pink rectangles.

The setK(m) is the non-escaping locus underF : s

t

7→d s

t

.

Combinatorics of primitive majors (work of Tomasini)

d=2 d=3

d=4

J. Tomasini Énumération des PM de degn

Combinatorics of primitive majors (work of Tomasini)

d=5

J. Tomasini Énumération des PM de degn

The total number p ˜

is

d−1

X

k=0

˜

p

(by Tomasini, ’2013), where

p_d^k = 1 d

d−1

X

i=k

i

k d −1+i

i d

d −1−i

(−1)^d−1−i

˜p_d^k = 1 d +k−1











p_d^k+ X

`≥2

`|d−1

`|k

ϕ(`)d−1

` +1

p<sub>(d−1)/`+1</sub><sup>k/`</sup> +

X

`≥2`|d

`|k−1

ϕ(`)

k−1

` +1

p<sub>d/`</sub><sup>(k−1)/`+1</sup>+ d +k−1

`

p<sub>d/`</sub><sup>(k−1)/`</sup>











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

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

Thurston’s mathematica animations

Q 3 θ 7→ leading eigenvalue of m

, plot of Thurston

Is this function continuous ?

Monotonicity and continuity along the veins

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

See works of Li and Tiozzo ...

Thank you ! Merci !

谢谢!