On decomposition of factor maps between shift spaces on groups - Z to - - PowerPoint PPT Presentation

On decomposition of factor maps between shift spaces

• n groups - Z to countable amenable groups

(joint with Kevin McGoff (Charlotte, NC) and Ronnie Pavlov (Denver))

Uijin Jung

Ajou University, South Korea

Minisymposium on discrete dynamical systems, Shanghai Jiao Tong University

Decompositions of codes on groups (U.Jung) 1

Contents

Introduction Decompositions of factor codes - on Z Decompositions of factor codes - on countable amenable group

Decompositions of codes on groups (U.Jung) Introduction 2

Shift spaces and codes

◮ The full A-shift AZ is the set of all bi-infinite sequences over a finite set A. ◮ The shift map σ on AZ is defined by σ(x)i = xi+1. A shift space, or a subshift is a σ-invariant closed subset of a full shift. ◮ A sliding block code (simply, a code) is a σ-commuting continuous map between shift spaces; φ is a factor code (conjugacy) if it is surjective (bijective). ◮ A topological Markov chain determined by an r × r, 0-1 matrix A is the set

• f all x = (xi) ∈ {1, . . . , r}Z with Axixi+1 = 1 for i ∈ Z.

◮ A subshift is an SFT if it is conjugate to a topological Markov chain. ◮ A sofic shift is an image of an SFT under a code.

Decompositions of codes on groups (U.Jung) Introduction 3

Decomposition of codes

◮ A decomposition of a code φ is a tuple (φ1, · · · , φn) of codes between shift spaces such that φ = φn · · · φ1.

Theorem (Williams, Nasu)

Every conjugacy between two shift spaces is the composition of simple elementary conjugacies, namely, splitting codes and amalgamation codes. Decompositions of codes arise in automorphism groups of SFTs, (eventual) factor theorems, lifting factor maps to closing maps, and construction of SFTs between two shifts, and so on.

Decompositions of codes on groups (U.Jung) Introduction 4

Decomposition Problems

Question (Adler and Marcus)

Can any factor code between irreducible SFTs with the same entropy be represented as a composition of closing codes? answered negatively by Kitchens. However,

Theorem (Kitchens, Marcus and Trow)

Let φ : X → Y be a factor code between irreducible SFTs with the same entropy. Then for all large n ∈ N, the code φ : Xn → Y n is a composition of closing codes.

Theorem (Kitchens, Marcus and Trow; Boyle)

Let φ : X → Y be a factor code between irreducible SFTs with the same entropy. Then there is a factor code ψ : Z → X such that φ ◦ ψ is a composition of closing codes.

Decompositions of codes on groups (U.Jung) Introduction 5

Decomposition Problems

Question (Adler and Marcus)

Can any factor code between irreducible SFTs with the same entropy be represented as a composition of closing codes?

Question (Trow)

Can any factor code between irreducible SFTs with the same entropy be decomposed only in (essentially) finitely many different ways?

Theorem (Boyle)

Let φ be a factor code between irreducible SFTs with the same entropy. Then the number of conjugacy classes of decompositions of φ is finite. Question: What happens if the entropies are different?

Decompositions of codes on groups (U.Jung) Introduction 6

Lindenstrauss’ Theorem

Question: What happens if the entropies are different?

Theorem (Lindenstrauss)

Let φ : (X, T) → (Y, S) be a factor between topological dynamical systems with X and Y of finite dimension. Let h ∈ [h(S), h(T)]. Then there are a system (Z, U) and factors φ1 : (X, T) → (Z, U) and φ2 : (Z, U) → (Y, S) such that φ = φ2 ◦ φ1 and h(U) = h. Even when X and Y are shifts of finite type, constructed Z is far from a subshift. We want to find intermediate systems as ”subshifts” rather than just topological dynamical systems.

Decompositions of codes on groups (U.Jung) Introduction 7

Contents

Introduction Decompositions of factor codes - on Z Decompositions of factor codes - on countable amenable group

Decompositions of codes on groups (U.Jung) Decompositions of factor codes - on Z 8

The set of the entropies of intermediate shifts

Let φ : X → Y be a factor code between subshifts with h(X) > h(Y ). ◮ Let S(φ) = {h(φ1(X)) : φ = φ2φ1 with φ1, φ2 factor codes }. ◮ Let S0(φ) = {h(φ1(X)) : φ = φ2φ1 with φ1, φ2 factors with φ1(X) SFT}.

Proposition (Boyle and Tuncel)

Let X and Y be irreducible SFTs. Then every element in S0(φ) \ {h(Y )} is a limit point of S0(φ).

Corollary

Let X and Y be irreducible SFTs. Then the number of conjugacy classes of decompositions is infinite.

Proposition (Boyle and Tuncel)

Let X and Y be irreducible SFTs. Then every element in S0(φ) \ {h(Y )} is a limit point of S0(φ).

Theorem (Hong, J. and Lee)

Let X be an SFT. Then S0(φ) is dense in [h(Y ), h(X)]. We conjecture that if X is mixing, then S (φ) = [h(Y ), h(X)] ∩ {log Perron }.

Decompositions of codes on groups (U.Jung) Decompositions of factor codes - on Z 9

Density result holds for any Z-subshift

Theorem (J, McGoff and Pavlov)

Let φ : X → Y be a factor code. Then S(φ) is dense in [h(Y ), h(X)].

Corollary (J, McGoff and Pavlov)

Let X be a Z-subshift. Then the set of the entropies of the subshift factors of X is dense in [0, h(X)].

Corollary (J, McGoff and Pavlov)

Let X be a Z-subshift. Then the set of the entropies of the 0-dimensional TDS factors of X is precisely [0, h(X)]. ◮ Given h ∈ (0, h(X)), one can construct a chain of subshifts {Xi}i∈Z with h(Xi) → h such that X → Xi+1 → Xi is a decomposition of X → Xi. ◮ The inverse limit of the system {Xi}i∈Z is a 0-dimensional TDS factor with entropy h.

Decompositions of codes on groups (U.Jung) Decompositions of factor codes - on Z 10

Main theorem and the Marker Lemma

Theorem (J, McGoff and Pavlov)

Let φ : X → Y be a factor code. Then S(φ) is dense in [h(Y ), h(X)]. The following Marker Lemma is an essential ingredient of the proof.

Lemma (Krieger)

Let X be a subshift and N ≥ 1. Then there is a clopen set F ⊂ X such that

• 1. σi(F), 0 ≤ i < N, are disjoint.
• 2. if σi(x) /

∈ F for −N < i < N, then x[−N,N] is p-periodic for some p < N. One can ‘mark’ the i-coordinate of x if σi(x) ∈ F. If two marked coordinates are far, then the intermediate part looks like a periodic point.

Decompositions of codes on groups (U.Jung) Decompositions of factor codes - on Z 11

Sketch of the proof

Theorem (J, McGoff and Pavlov)

Let φ : X → Y be a factor code. Then S(φ) is dense in [h(Y ), h(X)]. ◮ Assume φ is 1-block and AX ∩ AY = ∅. Take a large N ∈ N and choose a clopen “marker” set F ⊂ X. ◮ Define 1-block codes fN,k : X → (AX ∪ AY )Z by fN,k(x)0 =

• φ(x0)

if σj(x) ∈ F for some 0 ≤ j < k x0

• therwise

and with the commuting property. Intuitively, fN,k applies φ to the k-letters to the left of every occurrence of a marker coordinate of x.

Decompositions of codes on groups (U.Jung) Decompositions of factor codes - on Z 12

Sketch of the proof, II

Theorem (J, McGoff and Pavlov)

Let φ : X → Y be a factor code. Then S(φ) is dense in [h(Y ), h(X)]. ◮ Let Zk be the subshift consisting of all points over AX ∪ AY such that no word of the form awb with a, b ∈ AX and w ∈ k

j=1 Aj Y occur.

◮ Then fN,k(X) ⊂ Zk. ◮ The

n∈N Zn has the nonwandering set contained in P ∪ Y , where

P = {x ∈ X : x is periodic with period < N}. So limn h(fN,k(X)) = h(Y ). ◮ The difference between fN,k(X) and fN,k+1(X) is: the letters exactly k + 1 to the left of each marker symbol are mapped by φ. The frequency of such change is less than 1/N. ◮ Hence the set of the entropies h(fN,k(X)) is log |A|

N

• dense in [h(Y ), h(X)].

◮ As N → ∞, we are done.

Decompositions of codes on groups (U.Jung) Decompositions of factor codes - on Z 13

Contents

Introduction Decompositions of factor codes - on Z Decompositions of factor codes - on countable amenable group

Decompositions of codes on groups (U.Jung) Decompositions of factor codes - on countable amenable group 14

Density result holds for any G-subshift

Let G be a countable amenable group and X, Y be subshifts on G.

Theorem (J, McGoff and Pavlov)

Let φ : X → Y be a factor code. Then S(φ) is dense in [h(Y ), h(X)].

Corollary

Let X be a G-subshift. Then the set of the entropies of the subshift factors of X is dense in [0, h(X)].

Corollary

Let X be a G-subshift. Then the set of the entropies of the 0-dimensional G-TDS factors of X is precisely [0, h(X)].

Decompositions of codes on groups (U.Jung) Decompositions of factor codes - on countable amenable group 15

Shift spaces and sliding block codes over groups

Let A be a finite set and G be a group. As in Z-case... ◮ The full A-shift on G AG = {x : G → A} consists of all functions (configurations) from G to A.

◮ We use both xg and x(g) for the symbol at position g ∈ G.

◮ For each g ∈ G, let σg : AG → AG be the map (σg(x))h = xg−1h. The shift action on G × AZ is defined by σ(g, x)h = (σg(x))h = xg−1h. ◮ A shift space on G, or a subshift is a subset X ⊂ AG which is closed and σ-invariant (that is, σg(X) ⊂ X for each g ∈ G). ◮ A sliding block code is a σ-commuting continuous map between shift spaces: σg ◦ φ = φ ◦ σg for each g ∈ G.

Decompositions of codes on groups (U.Jung) Decompositions of factor codes - on countable amenable group 16

(Countable) amenable groups

Let G be a countable group. ◮ We say that G is amenable if there are finite (F¨

• lner) sets Fn ⊂ G such that

lim

n→∞

|gFn△Fn| |Fn| = 0 for each g ∈ G. ◮ For a subshift X ⊂ AG on an amenable group with {Fn}, the topological entropy can be defined by h(X) = limn

1 |Fn||BFn(X)|, where BF (X) is the

set of X-patterns on a set F ⊂ G.

◮ Entropy theory goes nicely to subshifts and their factors on an amenable group.

◮ We say A ⊂ G is (K, δ)-invariant if |{g ∈ G : Kg ∩ A = ∅ and Kg ∩ (G \ A) = ∅}| < δ|A|.

◮ that is, ‘K-boundary’ of A is δ-portion small. ◮ If A is (Fn, δ) small and Fn is F¨

• lner, then we can estimate the number of

patterns on A using the topological entropy: BA(X) ∼ e|A|(h(X)+ǫ).

Decompositions of codes on groups (U.Jung) Decompositions of factor codes - on countable amenable group 17

(Countable) amenable groups

Let G be a countable amenable group with a F¨

• lner sequence {Fn}.

◮ We say {T1, · · · , Tn} ǫ-quasi-tile a group G if {e} ⊂ T1 ⊂ · · · ⊂ Tn and, for any finite D ⊂ G there are Ci such that

• 1. for fixed i, {Tic : c ∈ Ci} are ǫ-disjoint
• 2. for i = j, TiCi ∩ TjCj = ∅
• 3. the collection {TiCi} (1 − ǫ)-cover D (i.e., |D \ TiCi| < ǫ|D|)

Theorem (Ornstein and Weiss)

Let ǫ > 0. There is N such that in any amenable group G and for any (K, δ), there are sets {T1, · · · , TN} that are (K, δ)-invariant sets and ǫ-quasi-tile G.

Decompositions of codes on groups (U.Jung) Decompositions of factor codes - on countable amenable group 18

Marker Lemma for G

Lemma

Let X be a G-subshift. For any finite sets S, T ⊂ G and c ∈ N, there is a clopen set F ⊂ X such that

• 1. For any c distinct elements s1, · · · , sc ∈ S, the sets {σsi(F)}c

i=1 have empty

intersection.

• 2. if x /

∈

i/ ∈S·S−1 σiF, then x(T) has at least c (essentially disjoint) periods in

SS−1. ◮ Krieger’s original marker lemma corresponds to G = Z, c = 2, S = [0, N] and T = [−N, N]. ◮ The first part guarantees “low density of visits to the set F”, and the second part guarantees “small entropy”.

Decompositions of codes on groups (U.Jung) Decompositions of factor codes - on countable amenable group 19

Sketch of the proof

Theorem (J, McGoff and Pavlov)

Let G be a countable amenable group and X, Y be subshifts on G. Let φ : X → Y be a factor code. Then S(φ) is dense in [h(Y ), h(X)]. ◮ Let ǫ, δ small, and c large. Find a set {Si} which δ-quasi tiles G and mini |Si| large enough. Let S = Si. ◮ For G = {g1, g2, g3, · · · } and a clopen set F, define 1-block codes fF,k : X → (AX ∪ AY )Z by x ∈ F ⇒ fF,n(x)g =

• φ(xg)

if g ∈ {g1, · · · , gn} xg

• therwise

◮ Then one can show that for any F and n ∈ N, h(fF,n(X)) − h(fF,n+1(X)) ≤ log |A| · D(F), where D(F) satisfies D(F) → 0 as the upper Banach density of F goes 0.

Decompositions of codes on groups (U.Jung) Decompositions of factor codes - on countable amenable group 20

Sketch of the proof

Theorem (J, McGoff and Pavlov)

Let G be a countable amenable group and X, Y be subshifts on G. Let φ : X → Y be a factor code. Then S(φ) is dense in [h(Y ), h(X)]. ◮ By the first property of Marker Lemma and using a quasi-tiling of Fk by Si’s, the density of visits to F in Fk can be shown small enough. ◮ Hence no matter what T is, if {Si} δ-quasi-tiles G and c is large, then we have a small difference in the consecutive factors h(fF,n(X)). ◮ Let XF be the ‘limit’ of the system of fF,n(X). The theorem will be done if h(XF ) < h(Y ) + ǫ.

Decompositions of codes on groups (U.Jung) Decompositions of factor codes - on countable amenable group 21

Sketch of the proof

Theorem (J, McGoff and Pavlov)

Let G be a countable amenable group and X, Y be subshifts on G. Let φ : X → Y be a factor code. Then S(φ) is dense in [h(Y ), h(X)]. ◮ Choose another tilings {T1, · · · , Tn} which are (SS−1, δ)-invariant and ǫ-quasi tiles G. Letting T = Ti, we can now fix F from Marker Lemma. ◮ For each Fn in the F¨

• lner sequence, each XF pattern w on Fn is determined

by the product of

• 1. B(w) = {g ∈ Fn : w(g) ∈ AX} : small exponential growth by ergodic theorem
• 2. y(Fn) : the number of choices is ≤ e(h(Y )+ǫ)|Fn|.
• 3. w({g : w(g) ∈ AX}) given B(w) : small exponential growth by periodicity

and quasi-tiling property of Si’s.

◮ And one can obtain h(Y ) − h(XF ) is small enough.

Decompositions of codes on groups (U.Jung) Decompositions of factor codes - on countable amenable group 22