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

On decomposition of factor maps between shift spaces on 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 A Z is the set of all bi-infinite sequences over a finite set A . ◮ The shift map σ on A Z is defined by σ ( x ) i = x i +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 of all x = ( x i ) ∈ { 1 , . . . , r } Z with A x i x i +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 ( φ 1 , · · · , φ n ) of codes between shift spaces such that φ = φ n · · · φ 1 . 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 : X n → 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 φ 1 : ( X, T ) → ( Z, U ) and φ 2 : ( Z, U ) → ( Y, S ) such that φ = φ 2 ◦ φ 1 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 ( φ 1 ( X )) : φ = φ 2 φ 1 with φ 1 , φ 2 factor codes } . ◮ Let S 0 ( φ ) = { h ( φ 1 ( X )) : φ = φ 2 φ 1 with φ 1 , φ 2 factors with φ 1 ( X ) SFT } . Proposition (Boyle and Tuncel) Let X and Y be irreducible SFTs. Then every element in S 0 ( φ ) \ { h ( Y ) } is a limit point of S 0 ( 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 S 0 ( φ ) \ { h ( Y ) } is a limit point of S 0 ( φ ) . Theorem (Hong, J. and Lee) Let X be an SFT. Then S 0 ( φ ) is dense in [ h ( Y ) , h ( X )] . Decompositions of codes on groups (U.Jung) Decompositions of factor codes - on Z 9 We conjecture that if X is mixing, then S ( φ ) = [ h ( Y ) , h ( X )] ∩ { log Perron } .

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 { X i } i ∈ Z with h ( X i ) → h such that X → X i +1 → X i is a decomposition of X → X i . ◮ The inverse limit of the system { X i } 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. σ i ( F ) , 0 ≤ i < N , are disjoint. 2. if σ i ( 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 σ i ( 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 A X ∩ A Y = ∅ . Take a large N ∈ N and choose a clopen “marker” set F ⊂ X . ◮ Define 1-block codes f N,k : X → ( A X ∪ A Y ) Z by � if σ j ( x ) ∈ F for some 0 ≤ j < k φ ( x 0 ) f N,k ( x ) 0 = x 0 otherwise and with the commuting property. Intuitively, f N,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 Z k be the subshift consisting of all points over A X ∪ A Y such that no word of the form awb with a, b ∈ A X and w ∈ � k j =1 A j Y occur. ◮ Then f N,k ( X ) ⊂ Z k . ◮ The � n ∈ N Z n has the nonwandering set contained in P ∪ Y , where P = { x ∈ X : x is periodic with period < N } . So lim n h ( f N,k ( X )) = h ( Y ) . ◮ The difference between f N,k ( X ) and f N,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 ( f N,k ( X )) is log | A | -dense in [ h ( Y ) , h ( X )] . N ◮ 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 factor codes - on countable Decompositions of codes on groups (U.Jung) 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 factor codes - on countable Decompositions of codes on groups (U.Jung) 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 A G = { x : G → A} consists of all functions (configurations) from G to A . ◮ We use both x g and x ( g ) for the symbol at position g ∈ G . ◮ For each g ∈ G , let σ g : A G → A G be the map ( σ g ( x )) h = x g − 1 h . The shift action on G × A Z is defined by σ ( g, x ) h = ( σ g ( x )) h = x g − 1 h . ◮ A shift space on G , or a subshift is a subset X ⊂ A G which is closed and σ -invariant (that is, σ g ( X ) ⊂ X for each g ∈ G ). ◮ A sliding block code is a σ -commuting continuous map between shift spaces: σ g ◦ φ = φ ◦ σ g for each g ∈ G . Decompositions of factor codes - on countable Decompositions of codes on groups (U.Jung) amenable group 16