Milnor-Thurston Kneading theory, ANR Marne-La-Vallee, 04-2014

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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)= ?

Def.1. If exists, Core(f) := Hubbard treeH, core-entropy:=h_top(f,H):=log(lap-growth).

Def.2. In all cases,

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 ?

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What is a core and its entropy for a polynomial ?

Def.1. If exists,Core(f) := Hubbard treeH, core-entropy:=h_top(f,H):=log(lap-growth).

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What is a core and its entropy for a polynomial ?

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What is a core and its entropy for a polynomial ?

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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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Lemma : When H exists, e

^h^top^(f^,H)

= d

dim(Bi-acc(f))

The idea came from Milnor-Thurston’s study on the real quadratic family

f_c(z) :=z²+c c ∈[−2,1

4], h(c) :=h_top(f_c,[c,c²+c]) Theorem.c 7→s(c) =exp(h(c))iscontinuous(Milnor-Thurston) andweakly decreasing (Douady), 2&1.

For the monotonicity, complex methods using external rays are more powerful...

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Lemma : When H exists, e

^h^top^(f^,H)

= d

dim(Bi-acc(f))

The idea came from Milnor-Thurston’s study on the real quadratic family

f_c(z) :=z²+c c ∈[−2,1

4], h(c) :=h_top(f_c,[c,c²+c]) Theorem.c 7→s(c) =exp(h(c))iscontinuous(Milnor-Thurston) andweakly decreasing (Douady), 2&1.

For the monotonicity, complex methods using external rays are more powerful...

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(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).

When−2←c,Yc %, so h(c)%. A single function, various subsets...

airplane1.png

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W. Thurston’s torus model, 2011

airplane1.png

1. A single map : F :

s t

7→2 s

t

,T²→T². Ac :={(t,1−t)∈T²,t∈Yc},

= bi-angles landing at a common point on R, is closed forward invariant set, with the same entropy.

2. Connectentropyto dimension:

2^dimA^c ^Thurston= eCore-entropy(fc) Tiozzo

= 2^dim^Y^c .

3. Bi-acc(fc):=bi-angles landing at a common point on the Julia set, has A_c as an attractor, and

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 torus model, 2011

airplane1.png

s t

7→2 s

t

,T²→T². Ac :={(t,1−t)∈T²,t∈Yc},

= 2^dim^Y^c .

= 2^dim^Y^c.

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W. Thurston’s torus model, 2011

airplane1.png

s t

7→2 s

t

,T²→T². Ac :={(t,1−t)∈T²,t∈Yc},

= 2^dim^Y^c .

= 2^dim^Y^c.

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Combinatorial approach of W. Thurston

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 Γ_m s.t. λ(Γ_m) =Growth(m)

• Parameter space

• For anym, Growth(m) =exph

T²(F,B(m))in Bowen’s sense

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Combinatorial approach of W. Thurston

s t

mod 17→

d·s d·t

To define

• m,B(m) then

Results :

• Parameter space

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Combinatorial approach of W. Thurston

s t

mod 17→

d·s d·t

To define

• m,B(m) then

Results :

• Parameter space

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A primitive major(critical portrait) mof degree d =

{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 major(critical portrait) mof degree d =

{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 ←→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.

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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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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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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 inside of

M={c ∈C : f_c^◦n(0)9∞}

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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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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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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 !

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Relate to Milnor-Thurston’s definition

Forn≥0, θ(fⁿ|_x+) :=sgn(fⁿ)⁰|_x+ (f⁰=id). Generating function (κA, κB, κC)_x⁺(t) := X

fⁿ(x⁺)∈A

θ(fⁿ|_x⁺)tⁿ ,inB,inC

. (1−s_At)κ_A = X

fⁿ(x⁺)∈A

θ(fⁿ|_x⁺)tⁿ− X

fⁿ(x⁺)∈A

θ(fⁿ⁺¹|_x⁺)tⁿ⁺¹

etc., adding the three : (κ_A, κ_B, κ_C)

1−sAt 1−sBt 1−sCt

!

=X

n≥0

−X

n≥1

=1.

knead. matrix N(t) = (˜A,B˜,C˜) = (κA, κB, κC)_c⁺

1 −(κA, κB, κC)_c⁻

1

(κ_A, κ_B, κ_C)_c⁺

2 −(κ_A, κ_B, κ_C)_c⁻

2

! .

Above (kernel vector)+Cramer gives

|B˜C˜|

1−s_At =− |A˜C˜|

1−s_Bt = |A˜B|˜

1−s_Ct =kneading determinant D(t).

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D

_old

= D

_new

in the unimodal case, with s

_A

< 0, s

_B

> 0

N(t) = (D_B,D_A) := (−1,+1) + ( X

n≥1,fⁿ(0^±)∈A

±2tⁿ, X

n≥1,fⁿ(0^±)∈B

±2tⁿ)

with+iffⁿ has a local minimum and−if local maximum. Then D_B =−(1−s_Bt)Dnew,D_A= (1−s_At)Dnew.

Sinceθn=sgn(fⁿ⁺¹)⁰ =s_interval·sgn(fⁿ)⁰, 2tD_old = (D_B,D_A)

s_At s_Bt

kernel vec.

= (D_B,D_A) 1

1

= (−1+s_Bt+1−s_At)D_new =2tD_new. Or 1

2R(t) = 1

2(D_A−D_B) = 1

2 D_B,D_A s_A

s_B

=D_old

=1+X

n≥1

±ε(n)tⁿ with ε(n) =

1 iffⁿ(0^±)>0

−1 iffⁿ(0^±)<0 . Clearly ε(n) determines whetherf changes the type of extrema of fⁿ(0).

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Journées dynamiques holomorphes, 28-29 Mai, Angers

Confirmed speakers : R. Dujardin, N. Mihalache, P. Roesch, D.Thurston, G. Tiozzo, J. Tomasini