Sofic entropy theory Lewis Bowen Logic, Dynamics and their - PowerPoint PPT Presentation

Sofic entropy theory Lewis Bowen Logic, Dynamics and their Interactions with a celebration of the work of Dan Mauldin University of North Texas June 2012 Lewis Bowen (Texas A&M) Sofic entropy theory 1 / 29

Group actions Let G be a countable discrete group. Lewis Bowen (Texas A&M) Sofic entropy theory 2 / 29

Group actions Let G be a countable discrete group. Let X be a set with a measure µ satisfying µ ( X ) = 1. Lewis Bowen (Texas A&M) Sofic entropy theory 2 / 29

Group actions Let G be a countable discrete group. Let X be a set with a measure µ satisfying µ ( X ) = 1. Let T : G → Aut ( X , µ ) be a homomorphism. Lewis Bowen (Texas A&M) Sofic entropy theory 2 / 29

Group actions Let G be a countable discrete group. Let X be a set with a measure µ satisfying µ ( X ) = 1. Let T : G → Aut ( X , µ ) be a homomorphism. T is a (probability-measure-preserving) action of G . Lewis Bowen (Texas A&M) Sofic entropy theory 2 / 29

Group actions Let G be a countable discrete group. Let X be a set with a measure µ satisfying µ ( X ) = 1. Let T : G → Aut ( X , µ ) be a homomorphism. T is a (probability-measure-preserving) action of G . Two actions T 1 : G → Aut ( X 1 , µ 1 ) , T 2 : G → Aut ( X 2 , µ 2 ) are isomorphic if there exists a measure-space isomorphism φ : X 1 → X 2 with φ ( T 1 ( g ) x ) = T 2 ( g ) φ ( x ) for a.e. x ∈ X 1 , for every g ∈ G . Lewis Bowen (Texas A&M) Sofic entropy theory 2 / 29

Group actions Let G be a countable discrete group. Let X be a set with a measure µ satisfying µ ( X ) = 1. Let T : G → Aut ( X , µ ) be a homomorphism. T is a (probability-measure-preserving) action of G . Two actions T 1 : G → Aut ( X 1 , µ 1 ) , T 2 : G → Aut ( X 2 , µ 2 ) are isomorphic if there exists a measure-space isomorphism φ : X 1 → X 2 with φ ( T 1 ( g ) x ) = T 2 ( g ) φ ( x ) for a.e. x ∈ X 1 , for every g ∈ G . Main Problem: Classify actions of G up to isomorphism. Lewis Bowen (Texas A&M) Sofic entropy theory 2 / 29

Bernoulli shifts Let ( K , κ ) be a probability space. Lewis Bowen (Texas A&M) Sofic entropy theory 3 / 29

Bernoulli shifts Let ( K , κ ) be a probability space. G acts on K G := { x : G → K } by shifting: ( gx )( f ) = x ( g − 1 f ) ∀ x ∈ K G , g , f ∈ G . Lewis Bowen (Texas A&M) Sofic entropy theory 3 / 29

Bernoulli shifts Let ( K , κ ) be a probability space. G acts on K G := { x : G → K } by shifting: ( gx )( f ) = x ( g − 1 f ) ∀ x ∈ K G , g , f ∈ G . The action G � ( K G , κ G ) = ( K , κ ) G is the Bernoulli shift over G with base measure κ . Lewis Bowen (Texas A&M) Sofic entropy theory 3 / 29

Bernoulli shifts Let ( K , κ ) be a probability space. G acts on K G := { x : G → K } by shifting: ( gx )( f ) = x ( g − 1 f ) ∀ x ∈ K G , g , f ∈ G . The action G � ( K G , κ G ) = ( K , κ ) G is the Bernoulli shift over G with base measure κ . � � H ( K , κ ) := − � k ∈ K ′ κ ( k ) log κ ( k ) if κ is supported on a countable set K ′ ⊂ K . Otherwise, H ( K , κ ) := + ∞ . Lewis Bowen (Texas A&M) Sofic entropy theory 3 / 29

Classification of Bernoulli Shifts G � ( K , κ ) G ∼ ? = G � ( L , λ ) G ⇐ ⇒ H ( K , κ ) = H ( L , λ ) Lewis Bowen (Texas A&M) Sofic entropy theory 4 / 29

Classification of Bernoulli Shifts G � ( K , κ ) G ∼ ? = G � ( L , λ ) G ⇐ ⇒ H ( K , κ ) = H ( L , λ ) = ⇒ G = Z , Kolmogorov (1958) Lewis Bowen (Texas A&M) Sofic entropy theory 4 / 29

Classification of Bernoulli Shifts G � ( K , κ ) G ∼ ? = G � ( L , λ ) G ⇐ ⇒ H ( K , κ ) = H ( L , λ ) = ⇒ G = Z , Kolmogorov (1958) ⇒ = G amenable, folk theorem (1970s) Lewis Bowen (Texas A&M) Sofic entropy theory 4 / 29

Classification of Bernoulli Shifts G � ( K , κ ) G ∼ ? = G � ( L , λ ) G ⇐ ⇒ H ( K , κ ) = H ( L , λ ) = ⇒ G = Z , Kolmogorov (1958) ⇒ = G amenable, folk theorem (1970s) Lewis Bowen (Texas A&M) Sofic entropy theory 4 / 29

Classification of Bernoulli Shifts G � ( K , κ ) G ∼ ? = G � ( L , λ ) G ⇐ ⇒ H ( K , κ ) = H ( L , λ ) = ⇒ G = Z , Kolmogorov (1958) ⇒ = G amenable, folk theorem (1970s) ⇐ = G = Z , Ornstein (1970) Lewis Bowen (Texas A&M) Sofic entropy theory 4 / 29

Classification of Bernoulli Shifts G � ( K , κ ) G ∼ ? = G � ( L , λ ) G ⇐ ⇒ H ( K , κ ) = H ( L , λ ) = ⇒ G = Z , Kolmogorov (1958) ⇒ = G amenable, folk theorem (1970s) ⇐ = G = Z , Ornstein (1970) ⇐ = G ⊃ Ornstein subgroup, Stepin (1975) Lewis Bowen (Texas A&M) Sofic entropy theory 4 / 29

Classification of Bernoulli Shifts G � ( K , κ ) G ∼ ? = G � ( L , λ ) G ⇐ ⇒ H ( K , κ ) = H ( L , λ ) = ⇒ G = Z , Kolmogorov (1958) ⇒ = G amenable, folk theorem (1970s) ⇐ = G = Z , Ornstein (1970) ⇐ = G ⊃ Ornstein subgroup, Stepin (1975) ⇐ | G | = ∞ & G amenable, = Ornstein-Weiss (1987) Lewis Bowen (Texas A&M) Sofic entropy theory 4 / 29

Classification of Bernoulli Shifts G � ( K , κ ) G ∼ ? = G � ( L , λ ) G ⇐ ⇒ H ( K , κ ) = H ( L , λ ) = ⇒ G = Z , Kolmogorov (1958) ⇒ = G amenable, folk theorem (1970s) ⇐ = G = Z , Ornstein (1970) ⇐ = G ⊃ Ornstein subgroup, Stepin (1975) ⇐ | G | = ∞ & G amenable, = Ornstein-Weiss (1987) ⇐ = | G | = ∞ , ( K , κ ) , ( L , λ ) not 2-atom spaces, B. (2011) Lewis Bowen (Texas A&M) Sofic entropy theory 4 / 29

Classification of Bernoulli Shifts G � ( K , κ ) G ∼ ? = G � ( L , λ ) G ⇐ ⇒ H ( K , κ ) = H ( L , λ ) = ⇒ G = Z , Kolmogorov (1958) ⇒ = G amenable, folk theorem (1970s) = ⇒ G sofic, B. (2010) ⇐ = G = Z , Ornstein (1970) ⇐ G ⊃ Ornstein subgroup, Stepin (1975) = ⇐ = | G | = ∞ & G amenable, Ornstein-Weiss (1987) ⇐ = | G | = ∞ , ( K , κ ) , ( L , λ ) not 2-atom spaces, B. (2011) Lewis Bowen (Texas A&M) Sofic entropy theory 5 / 29

Factor maps Lewis Bowen (Texas A&M) Sofic entropy theory 6 / 29

Factor maps Definition Let G � ( X , µ ) , G � ( Y , ν ) be two systems and φ : X → Y a measurable map with φ ∗ µ = ν , φ ( gx ) = g φ ( x ) for a.e. x ∈ X and all g ∈ G . Then φ is a factor map from G � ( X , µ ) to G � ( Y , ν ) . Lewis Bowen (Texas A&M) Sofic entropy theory 6 / 29

Factor maps Definition Let G � ( X , µ ) , G � ( Y , ν ) be two systems and φ : X → Y a measurable map with φ ∗ µ = ν , φ ( gx ) = g φ ( x ) for a.e. x ∈ X and all g ∈ G . Then φ is a factor map from G � ( X , µ ) to G � ( Y , ν ) . Question: which Bernoulli shifts factor onto which Bernoulli shifts? Lewis Bowen (Texas A&M) Sofic entropy theory 6 / 29

Factor maps Definition Let G � ( X , µ ) , G � ( Y , ν ) be two systems and φ : X → Y a measurable map with φ ∗ µ = ν , φ ( gx ) = g φ ( x ) for a.e. x ∈ X and all g ∈ G . Then φ is a factor map from G � ( X , µ ) to G � ( Y , ν ) . Question: which Bernoulli shifts factor onto which Bernoulli shifts? Theorem (Sinai 1960s, Ornstein-Weiss 1980) If G is amenable then G � ( K , κ ) G factors onto G � ( L , λ ) G if and only if H ( K , κ ) ≥ H ( L , λ ) . Lewis Bowen (Texas A&M) Sofic entropy theory 6 / 29

Factor maps Definition Let G � ( X , µ ) , G � ( Y , ν ) be two systems and φ : X → Y a measurable map with φ ∗ µ = ν , φ ( gx ) = g φ ( x ) for a.e. x ∈ X and all g ∈ G . Then φ is a factor map from G � ( X , µ ) to G � ( Y , ν ) . Question: which Bernoulli shifts factor onto which Bernoulli shifts? Theorem (Sinai 1960s, Ornstein-Weiss 1980) If G is amenable then G � ( K , κ ) G factors onto G � ( L , λ ) G if and only if H ( K , κ ) ≥ H ( L , λ ) . G amenable ⇒ h ( G � ( Z / n Z , u n ) G ) = log ( n ) (the full n -shift). So the full 2-shift over G cannot factor onto the full 4-shift over G . Lewis Bowen (Texas A&M) Sofic entropy theory 6 / 29

The Ornstein-Weiss Example Theorem (Ornstein-Weiss, 1987) If F = � a , b � is the rank 2 free group then the full 2 -shift over F factors onto the full 4 -shift over F . Lewis Bowen (Texas A&M) Sofic entropy theory 7 / 29

The Ornstein-Weiss Example Theorem (Ornstein-Weiss, 1987) If F = � a , b � is the rank 2 free group then the full 2 -shift over F factors onto the full 4 -shift over F . Define φ : ( Z / 2 Z ) F → ( Z / 2 Z × Z / 2 Z ) F by Lewis Bowen (Texas A&M) Sofic entropy theory 7 / 29

The Ornstein-Weiss Example Theorem (Ornstein-Weiss, 1987) If F = � a , b � is the rank 2 free group then the full 2 -shift over F factors onto the full 4 -shift over F . Define φ : ( Z / 2 Z ) F → ( Z / 2 Z × Z / 2 Z ) F by � � φ ( x )( g ) = x ( g ) + x ( ga ) , x ( g ) + x ( gb ) . Theorem (B. 2011) If G contains a subgroup isomorphic to F then every nontrivial Bernoulli shift over G factors onto every other Bernoulli shift over G. Lewis Bowen (Texas A&M) Sofic entropy theory 7 / 29

Factors between Bernoulli shifts Theorem (Gaboriau-Lyons, 2009) For every non-amenable group G there is some space ( K , κ ) with H ( K , κ ) < ∞ and an essentially free action F� ( K , κ ) G with F -orbits contained in the G-orbits. Lewis Bowen (Texas A&M) Sofic entropy theory 8 / 29