Entropy Theory for Sofic Group Actions Lewis Bowen Workshop on II 1 - PowerPoint PPT Presentation

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) Entropy Theory for Sofic Group Actions 9 / 48

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) Entropy Theory for Sofic Group Actions 9 / 48

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 ) . Lewis Bowen (Texas A&M) Entropy Theory for Sofic Group Actions 9 / 48

More on factors Open : If G is non-amenable, does every Bernoulli shift factor onto every Bernoulli shift? Lewis Bowen (Texas A&M) Entropy Theory for Sofic Group Actions 10 / 48

More on factors Open : If G is non-amenable, does every Bernoulli shift factor onto every Bernoulli shift? Ball (2005): for every non-amenable group G there is some m = m ( G ) > 0 such that the m -shift factors onto every Bernoulli shift. Bowen (2009): if G contains a rank 2 free subgroup then ‘yes’. Lewis Bowen (Texas A&M) Entropy Theory for Sofic Group Actions 10 / 48

Recap If G is non-amenable then, for some n , the n -shift factors onto all other Bernoulli shifts. Lewis Bowen (Texas A&M) Entropy Theory for Sofic Group Actions 11 / 48

Recap If G is non-amenable then, for some n , the n -shift factors onto all other Bernoulli shifts. So if “entropy theory” requires the n -shift to have entropy log ( n ) , and that entropy does not increase under factors then there is no entropy theory for non-amenable groups. Lewis Bowen (Texas A&M) Entropy Theory for Sofic Group Actions 11 / 48

New results Theorem (L. B., 2009) If G is a sofic group then Kolmogorov’s direction holds. I.e., if G � ( K G , κ G ) is isomorphic to G � ( L G , λ G ) then H ( K , κ ) = H ( L , λ ) . Lewis Bowen (Texas A&M) Entropy Theory for Sofic Group Actions 12 / 48

The case G = Z . Let T : X → X be an automorphism of ( X , µ ) . Lewis Bowen (Texas A&M) Entropy Theory for Sofic Group Actions 13 / 48

The case G = Z . Let T : X → X be an automorphism of ( X , µ ) . Let φ : X → A be an observable (i.e., a measurable map into a finite set). Lewis Bowen (Texas A&M) Entropy Theory for Sofic Group Actions 13 / 48

The case G = Z . Let T : X → X be an automorphism of ( X , µ ) . Let φ : X → A be an observable (i.e., a measurable map into a finite set). Let x ∈ X be a typical element and consider the sequence ( . . . , φ ( T − 1 x ) , φ ( x ) , φ ( Tx ) , . . . ) . Lewis Bowen (Texas A&M) Entropy Theory for Sofic Group Actions 13 / 48

The case G = Z . Let T : X → X be an automorphism of ( X , µ ) . Let φ : X → A be an observable (i.e., a measurable map into a finite set). Let x ∈ X be a typical element and consider the sequence ( . . . , φ ( T − 1 x ) , φ ( x ) , φ ( Tx ) , . . . ) . The idea: For n > 0, count the number of sequences ( a 1 , a 2 , . . . , a n ) with elements a i ∈ A that approximate the above sequence. Lewis Bowen (Texas A&M) Entropy Theory for Sofic Group Actions 13 / 48

Local statistics Let W ⊂ Z be finite. ( W stands for window ) Lewis Bowen (Texas A&M) Entropy Theory for Sofic Group Actions 14 / 48

Local statistics Let W ⊂ Z be finite. ( W stands for window ) Define φ W : X → A W = A × A × . . . × A by � �� � W � � φ W ( x ) := φ ( T w x ) w ∈ W . Lewis Bowen (Texas A&M) Entropy Theory for Sofic Group Actions 14 / 48

Local statistics Let W ⊂ Z be finite. ( W stands for window ) Define φ W : X → A W = A × A × . . . × A by � �� � W � � φ W ( x ) := φ ( T w x ) w ∈ W . ∗ µ is a measure on A W that encodes the local statistics . φ W Lewis Bowen (Texas A&M) Entropy Theory for Sofic Group Actions 14 / 48

Sequences Let ψ : { 1 , . . . , n } → A be a map. Lewis Bowen (Texas A&M) Entropy Theory for Sofic Group Actions 15 / 48

Sequences Let ψ : { 1 , . . . , n } → A be a map. ψ W : { 1 , . . . , n } → A W is defined by � � ψ W ( j ) = ψ ( j + w ) w ∈ W . Lewis Bowen (Texas A&M) Entropy Theory for Sofic Group Actions 15 / 48

Sequences Let ψ : { 1 , . . . , n } → A be a map. ψ W : { 1 , . . . , n } → A W is defined by � � ψ W ( j ) = ψ ( j + w ) w ∈ W . ∈ { 1 , . . . , n } ) (define it arbitrarily if j + w / Lewis Bowen (Texas A&M) Entropy Theory for Sofic Group Actions 15 / 48

Sequences Let ψ : { 1 , . . . , n } → A be a map. ψ W : { 1 , . . . , n } → A W is defined by � � ψ W ( j ) = ψ ( j + w ) w ∈ W . ∈ { 1 , . . . , n } ) (define it arbitrarily if j + w / Let u be the uniform measure on { 1 , . . . , n } . ψ W ∗ u is a measure on A W that encodes the local statistics of the sequence ( ψ ( 1 ) , . . . , ψ ( n )) ∈ A n . Lewis Bowen (Texas A&M) Entropy Theory for Sofic Group Actions 15 / 48

Entropy as a growth rate Let d W ( φ, ψ ) be the l 1 -distance between φ W ∗ µ and ψ W ∗ u : Lewis Bowen (Texas A&M) Entropy Theory for Sofic Group Actions 16 / 48

Entropy as a growth rate Let d W ( φ, ψ ) be the l 1 -distance between φ W ∗ µ and ψ W ∗ u : � � � � φ W ∗ µ ( α ) − ψ W � � d W ( φ, ψ ) := ∗ u ( α ) � . α ∈ A W Lewis Bowen (Texas A&M) Entropy Theory for Sofic Group Actions 16 / 48

Entropy as a growth rate Let d W ( φ, ψ ) be the l 1 -distance between φ W ∗ µ and ψ W ∗ u : � � � � φ W ∗ µ ( α ) − ψ W � � d W ( φ, ψ ) := ∗ u ( α ) � . α ∈ A W Theorem � �� 1 � � � h ( T , φ ) = inf W ⊂ Z inf ǫ> 0 lim n log ψ : { 1 , . . . , n } → A : d W ( φ, ψ ) < ǫ � . � n →∞ Lewis Bowen (Texas A&M) Entropy Theory for Sofic Group Actions 16 / 48

Sofic Groups A sofic approximation to G is a sequence Σ = { σ i } ∞ i = 1 of maps σ i : G → Sym ( m i ) such that for every f , g ∈ G , 1 1 lim |{ 1 ≤ p ≤ m i : σ i ( g ) σ i ( f ) p = σ i ( gf ) p }| = 1 m i i →∞ for every f � = g ∈ G , 2 1 |{ 1 ≤ p ≤ m i : σ i ( g ) p � = σ i ( f ) p }| = 1 lim m i i →∞ lim i →∞ m i = + ∞ . 3 Lewis Bowen (Texas A&M) Entropy Theory for Sofic Group Actions 17 / 48

Sofic Groups A sofic approximation to G is a sequence Σ = { σ i } ∞ i = 1 of maps σ i : G → Sym ( m i ) such that for every f , g ∈ G , 1 1 lim |{ 1 ≤ p ≤ m i : σ i ( g ) σ i ( f ) p = σ i ( gf ) p }| = 1 m i i →∞ for every f � = g ∈ G , 2 1 |{ 1 ≤ p ≤ m i : σ i ( g ) p � = σ i ( f ) p }| = 1 lim m i i →∞ lim i →∞ m i = + ∞ . 3 G is sofic if there exists a sofic approximation to G . Lewis Bowen (Texas A&M) Entropy Theory for Sofic Group Actions 17 / 48

Sofic Groups A sofic approximation to G is a sequence Σ = { σ i } ∞ i = 1 of maps σ i : G → Sym ( m i ) such that for every f , g ∈ G , 1 1 lim |{ 1 ≤ p ≤ m i : σ i ( g ) σ i ( f ) p = σ i ( gf ) p }| = 1 m i i →∞ for every f � = g ∈ G , 2 1 |{ 1 ≤ p ≤ m i : σ i ( g ) p � = σ i ( f ) p }| = 1 lim m i i →∞ lim i →∞ m i = + ∞ . 3 G is sofic if there exists a sofic approximation to G . Lewis Bowen (Texas A&M) Entropy Theory for Sofic Group Actions 17 / 48

Sofic Groups Residually finite groups are sofic. Hence all linear groups are sofic. Lewis Bowen (Texas A&M) Entropy Theory for Sofic Group Actions 18 / 48

Sofic Groups Residually finite groups are sofic. Hence all linear groups are sofic. Amenable groups are sofic. Lewis Bowen (Texas A&M) Entropy Theory for Sofic Group Actions 18 / 48

Sofic Groups Residually finite groups are sofic. Hence all linear groups are sofic. Amenable groups are sofic. (Gromov 1999, Weiss 2000, Elek-Szabo 2005) If G is sofic then G satisfies Gottshalk’s surjunctivity conjecture, Connes embedding conjecture, the Determinant conjecture, Kaplansky’s direct finiteness conjecture. Lewis Bowen (Texas A&M) Entropy Theory for Sofic Group Actions 18 / 48

Sofic Groups Residually finite groups are sofic. Hence all linear groups are sofic. Amenable groups are sofic. (Gromov 1999, Weiss 2000, Elek-Szabo 2005) If G is sofic then G satisfies Gottshalk’s surjunctivity conjecture, Connes embedding conjecture, the Determinant conjecture, Kaplansky’s direct finiteness conjecture. Open : Is every countable group sofic? Lewis Bowen (Texas A&M) Entropy Theory for Sofic Group Actions 18 / 48

Entropy for Sofic Groups Let G � ( X , µ ) be a system, Lewis Bowen (Texas A&M) Entropy Theory for Sofic Group Actions 19 / 48

Entropy for Sofic Groups Let G � ( X , µ ) be a system, Σ = { σ i } be a sofic approximation to G where σ i : G → Sym ( m i ) , Lewis Bowen (Texas A&M) Entropy Theory for Sofic Group Actions 19 / 48

Entropy for Sofic Groups Let G � ( X , µ ) be a system, Σ = { σ i } be a sofic approximation to G where σ i : G → Sym ( m i ) , φ : X → A be a measurable map into a finite set. Lewis Bowen (Texas A&M) Entropy Theory for Sofic Group Actions 19 / 48

Entropy for Sofic Groups Let G � ( X , µ ) be a system, Σ = { σ i } be a sofic approximation to G where σ i : G → Sym ( m i ) , φ : X → A be a measurable map into a finite set. The idea: Count the number of observables ψ : { 1 , . . . , m i } → A so that ( G , [ m i ] , u i , ψ ) approximates ( G , X , µ, φ ) . Lewis Bowen (Texas A&M) Entropy Theory for Sofic Group Actions 19 / 48

Approximating � �� If W ⊂ G is finite, let φ W : X → A W be the map φ W ( x ) := � φ wx w ∈ W . Lewis Bowen (Texas A&M) Entropy Theory for Sofic Group Actions 20 / 48

Approximating � �� If W ⊂ G is finite, let φ W : X → A W be the map φ W ( x ) := � φ wx w ∈ W . Given ψ : { 1 , . . . , m i } → A , ψ W : { 1 , . . . , m i } → A W is the map � �� � ψ W ( j ) := ψ σ ( w ) j w ∈ W . Lewis Bowen (Texas A&M) Entropy Theory for Sofic Group Actions 20 / 48

Approximating � �� If W ⊂ G is finite, let φ W : X → A W be the map φ W ( x ) := � φ wx w ∈ W . Given ψ : { 1 , . . . , m i } → A , ψ W : { 1 , . . . , m i } → A W is the map � �� � ψ W ( j ) := ψ σ ( w ) j w ∈ W . Let d W ( φ, ψ ) be the l 1 -distance between φ W ∗ µ and ψ W ∗ u . Lewis Bowen (Texas A&M) Entropy Theory for Sofic Group Actions 20 / 48

Entropy for sofic groups � � � { ψ : { 1 , . . . , m i } → A : d W ( φ, ψ ) ≤ ǫ } log � � � h Σ , φ := inf W ⊂ G inf ǫ> 0 lim sup . m i i →∞ Lewis Bowen (Texas A&M) Entropy Theory for Sofic Group Actions 21 / 48

Entropy for sofic groups � � � { ψ : { 1 , . . . , m i } → A : d W ( φ, ψ ) ≤ ǫ } log � � � h Σ , φ := inf W ⊂ G inf ǫ> 0 lim sup . m i i →∞ Theorem (L.B. ’09) � If φ 1 and φ 2 are generating then h Σ , φ 1 ) = h (Σ , φ 2 ) . So let h (Σ , G , X , µ ) be this common number. Lewis Bowen (Texas A&M) Entropy Theory for Sofic Group Actions 21 / 48

Entropy for sofic groups � � � { ψ : { 1 , . . . , m i } → A : d W ( φ, ψ ) ≤ ǫ } log � � � h Σ , φ := inf W ⊂ G inf ǫ> 0 lim sup . m i i →∞ Theorem (L.B. ’09) � If φ 1 and φ 2 are generating then h Σ , φ 1 ) = h (Σ , φ 2 ) . So let h (Σ , G , X , µ ) be this common number. Theorem (L.B. ’10, Kerr-Li ’10) � � If G is amenable then h Σ , G , X , µ is the classical entropy of ( G , X , µ ) . Lewis Bowen (Texas A&M) Entropy Theory for Sofic Group Actions 21 / 48

Entropy for sofic groups � � � { ψ : { 1 , . . . , m i } → A : d W ( φ, ψ ) ≤ ǫ } log � � � h Σ , φ := inf W ⊂ G inf ǫ> 0 lim sup . m i i →∞ Theorem (L.B. ’09) � If φ 1 and φ 2 are generating then h Σ , φ 1 ) = h (Σ , φ 2 ) . So let h (Σ , G , X , µ ) be this common number. Theorem (L.B. ’10, Kerr-Li ’10) � � If G is amenable then h Σ , G , X , µ is the classical entropy of ( G , X , µ ) . Theorem (L.B. ’09) � Σ , G , K G , κ G � h = H ( K , κ ) . Lewis Bowen (Texas A&M) Entropy Theory for Sofic Group Actions 21 / 48

Proof sketch Theorem � If φ 1 and φ 2 are generating then h Σ , φ 1 ) = h (Σ , φ 2 ) . So let h (Σ , G , X , µ ) be this common number. Lewis Bowen (Texas A&M) Entropy Theory for Sofic Group Actions 22 / 48

Proof sketch Theorem � If φ 1 and φ 2 are generating then h Σ , φ 1 ) = h (Σ , φ 2 ) . So let h (Σ , G , X , µ ) be this common number. Two observables φ : X → A , ψ : X → B are equivalent if the partitions { φ − 1 ( a ) : a ∈ A } , { ψ − 1 ( b ) : b ∈ B } agree up to measure zero. Lewis Bowen (Texas A&M) Entropy Theory for Sofic Group Actions 22 / 48

Proof sketch Theorem � If φ 1 and φ 2 are generating then h Σ , φ 1 ) = h (Σ , φ 2 ) . So let h (Σ , G , X , µ ) be this common number. Two observables φ : X → A , ψ : X → B are equivalent if the partitions { φ − 1 ( a ) : a ∈ A } , { ψ − 1 ( b ) : b ∈ B } agree up to measure zero. Let P be the set of all equivalence classes of observables φ with H ( φ ) < ∞ . Lewis Bowen (Texas A&M) Entropy Theory for Sofic Group Actions 22 / 48

Proof sketch Theorem � If φ 1 and φ 2 are generating then h Σ , φ 1 ) = h (Σ , φ 2 ) . So let h (Σ , G , X , µ ) be this common number. Two observables φ : X → A , ψ : X → B are equivalent if the partitions { φ − 1 ( a ) : a ∈ A } , { ψ − 1 ( b ) : b ∈ B } agree up to measure zero. Let P be the set of all equivalence classes of observables φ with H ( φ ) < ∞ . Definition (Rohlin distance) d ( φ, ψ ) := 2 H ( φ ∨ ψ ) − H ( ψ ) − H ( φ ) = H ( φ | ψ ) + H ( ψ | φ ) . Lewis Bowen (Texas A&M) Entropy Theory for Sofic Group Actions 22 / 48

Proof sketch Theorem � If φ 1 and φ 2 are generating then h Σ , φ 1 ) = h (Σ , φ 2 ) . So let h (Σ , G , X , µ ) be this common number. Two observables φ : X → A , ψ : X → B are equivalent if the partitions { φ − 1 ( a ) : a ∈ A } , { ψ − 1 ( b ) : b ∈ B } agree up to measure zero. Let P be the set of all equivalence classes of observables φ with H ( φ ) < ∞ . Definition (Rohlin distance) d ( φ, ψ ) := 2 H ( φ ∨ ψ ) − H ( ψ ) − H ( φ ) = H ( φ | ψ ) + H ( ψ | φ ) . Lewis Bowen (Texas A&M) Entropy Theory for Sofic Group Actions 22 / 48

Proof sketch Definition φ refines ψ if H ( ψ ∨ φ ) = H ( φ ) . Lewis Bowen (Texas A&M) Entropy Theory for Sofic Group Actions 23 / 48

Proof sketch Definition φ refines ψ if H ( ψ ∨ φ ) = H ( φ ) . Definition φ and ψ are combinatorially equivalent if there exists finite subsets K , L ⊂ G such that φ K refines ψ and ψ L refines φ . Lewis Bowen (Texas A&M) Entropy Theory for Sofic Group Actions 23 / 48

Proof sketch Theorem If φ is a generator then its combinatorial equivalence class is dense in the space of all generating observables. Lewis Bowen (Texas A&M) Entropy Theory for Sofic Group Actions 24 / 48

Proof sketch Theorem If φ is a generator then its combinatorial equivalence class is dense in the space of all generating observables. Lemma h (Σ , φ ) is upper semi-continuous in φ . Lewis Bowen (Texas A&M) Entropy Theory for Sofic Group Actions 24 / 48

Proof sketch Theorem If φ is a generator then its combinatorial equivalence class is dense in the space of all generating observables. Lemma h (Σ , φ ) is upper semi-continuous in φ . Theorem If φ and ψ are combinatorially equivalent then h (Σ , φ ) = h (Σ , ψ ) . Lewis Bowen (Texas A&M) Entropy Theory for Sofic Group Actions 24 / 48

Proof sketch Definition φ is a simple splitting of ψ if there exists f ∈ G and an observable ω refined by ψ such that φ = ψ ∨ ω ◦ f . φ is a splitting of ψ if it can be obtained from ψ by a sequence of simple splittings. Lewis Bowen (Texas A&M) Entropy Theory for Sofic Group Actions 25 / 48

Proof sketch Definition φ is a simple splitting of ψ if there exists f ∈ G and an observable ω refined by ψ such that φ = ψ ∨ ω ◦ f . φ is a splitting of ψ if it can be obtained from ψ by a sequence of simple splittings. Lemma If φ and ψ are equivalent then there exists an observable ω that is a splitting of both φ and ψ . Lewis Bowen (Texas A&M) Entropy Theory for Sofic Group Actions 25 / 48

Proof sketch Definition φ is a simple splitting of ψ if there exists f ∈ G and an observable ω refined by ψ such that φ = ψ ∨ ω ◦ f . φ is a splitting of ψ if it can be obtained from ψ by a sequence of simple splittings. Lemma If φ and ψ are equivalent then there exists an observable ω that is a splitting of both φ and ψ . Proposition If φ is a simple splitting of ψ then h (Σ , φ ) = h (Σ , ψ ) . Lewis Bowen (Texas A&M) Entropy Theory for Sofic Group Actions 25 / 48

Applications: von Neumann algebras Definition G 1 � ( X 1 , µ 1 ) and G 2 � ( X 2 , µ 2 ) are von Neumann equivalent (vNE) if L ∞ ( X 1 , µ 1 ) ⋊ G 1 ∼ = L ∞ ( X 2 , µ 2 ) ⋊ G 2 . Lewis Bowen (Texas A&M) Entropy Theory for Sofic Group Actions 26 / 48

Applications: von Neumann algebras Definition G 1 � ( X 1 , µ 1 ) and G 2 � ( X 2 , µ 2 ) are von Neumann equivalent (vNE) if L ∞ ( X 1 , µ 1 ) ⋊ G 1 ∼ = L ∞ ( X 2 , µ 2 ) ⋊ G 2 . Theorem (Popa 2006) If G is an ICC property (T) group then any two von Neumann equivalent Bernoulli shifts over G are isomorphic. Lewis Bowen (Texas A&M) Entropy Theory for Sofic Group Actions 26 / 48

Applications: von Neumann algebras Definition G 1 � ( X 1 , µ 1 ) and G 2 � ( X 2 , µ 2 ) are von Neumann equivalent (vNE) if L ∞ ( X 1 , µ 1 ) ⋊ G 1 ∼ = L ∞ ( X 2 , µ 2 ) ⋊ G 2 . Theorem (Popa 2006) If G is an ICC property (T) group then any two von Neumann equivalent Bernoulli shifts over G are isomorphic. Corollary If, in addition, G is sofic and Ornstein then Bernoulli shifts over G are classified up to vNE by base measure entropy. E.g., this occurs when G = PSL n ( Z ) for n > 2 . Lewis Bowen (Texas A&M) Entropy Theory for Sofic Group Actions 26 / 48

Applications: orbit equivalence Definition G 1 � ( X 1 , µ 1 ) is orbit equivalent (OE) to G 2 � ( X 2 , µ 2 ) if there exists a measure-space isomorphism φ : X 1 → X 2 such that φ ( G 1 x ) = G 2 φ ( x ) for a.e. x ∈ X 1 . Lewis Bowen (Texas A&M) Entropy Theory for Sofic Group Actions 27 / 48

Applications: orbit equivalence Definition G 1 � ( X 1 , µ 1 ) is orbit equivalent (OE) to G 2 � ( X 2 , µ 2 ) if there exists a measure-space isomorphism φ : X 1 → X 2 such that φ ( G 1 x ) = G 2 φ ( x ) for a.e. x ∈ X 1 . Theorem (Dye 1959, Ornstein-Weiss 1980) If G 1 and G 2 are amenable and infinite and their respective actions are ergodic and free then G 1 � ( X 1 , µ 1 ) is OE to G 2 � ( X 2 , µ 2 ) . Lewis Bowen (Texas A&M) Entropy Theory for Sofic Group Actions 27 / 48

OE rigidity Theorem For the following groups, OE of Bernoulli shift implies conjugacy of Bernoulli shifts: property (T) groups with ICC (Popa 2007), mapping class groups with 3 g + n − 4 > 0 , ( g , n ) / ∈ { ( 1 , 2 ) , ( 2 , 0 ) } (Kida, 2008), direct products of infinite non-amenable groups with no finite normal subgroups (Popa, 2008). Lewis Bowen (Texas A&M) Entropy Theory for Sofic Group Actions 28 / 48

OE rigidity Theorem For the following groups, OE of Bernoulli shift implies conjugacy of Bernoulli shifts: property (T) groups with ICC (Popa 2007), mapping class groups with 3 g + n − 4 > 0 , ( g , n ) / ∈ { ( 1 , 2 ) , ( 2 , 0 ) } (Kida, 2008), direct products of infinite non-amenable groups with no finite normal subgroups (Popa, 2008). Corollary If G is as above then Bernoulli shifts over G are classified up to OE by base measure entropy. Lewis Bowen (Texas A&M) Entropy Theory for Sofic Group Actions 28 / 48

Free Groups: a special case Let F = � s 1 , . . . , s r � . Let F act on ( X , µ ) . Lewis Bowen (Texas A&M) Entropy Theory for Sofic Group Actions 29 / 48

Free Groups: a special case Let F = � s 1 , . . . , s r � . Let F act on ( X , µ ) . Given an observable φ : X → A , define r � F ( φ ) := − ( 2 r − 1 ) H ( φ ) + H ( φ ∨ φ ◦ s i ); i = 1 � φ B ( e , n ) � f ( φ ) := inf n F . Lewis Bowen (Texas A&M) Entropy Theory for Sofic Group Actions 29 / 48

Free Groups: a special case Let F = � s 1 , . . . , s r � . Let F act on ( X , µ ) . Given an observable φ : X → A , define r � F ( φ ) := − ( 2 r − 1 ) H ( φ ) + H ( φ ∨ φ ◦ s i ); i = 1 � φ B ( e , n ) � f ( φ ) := inf n F . Theorem If φ 1 and φ 2 are generating then f ( φ 1 ) = f ( φ 2 ) . So we may define f ( F , X , µ ) = f ( φ 1 ) . Moreover, f ( F , K F , κ F ) = H ( K , κ ) . Lewis Bowen (Texas A&M) Entropy Theory for Sofic Group Actions 29 / 48

Free Groups: a special case For each n ≥ 1, let σ n : F = � s 1 , . . . , s r � → Sym ( n ) be chosen uniformly at random. Lewis Bowen (Texas A&M) Entropy Theory for Sofic Group Actions 30 / 48

Free Groups: a special case For each n ≥ 1, let σ n : F = � s 1 , . . . , s r � → Sym ( n ) be chosen uniformly at random. Define �� � � log E � { ψ : { 1 , . . . , n } → A : d W ( φ, ψ ) ≤ ǫ } � � � h ∗ φ := inf W inf ǫ> 0 lim sup . n n →∞ Lewis Bowen (Texas A&M) Entropy Theory for Sofic Group Actions 30 / 48

Free Groups: a special case For each n ≥ 1, let σ n : F = � s 1 , . . . , s r � → Sym ( n ) be chosen uniformly at random. Define �� � � log E � { ψ : { 1 , . . . , n } → A : d W ( φ, ψ ) ≤ ǫ } � � � h ∗ φ := inf W inf ǫ> 0 lim sup . n n →∞ Theorem � � h ∗ φ = f ( φ ) . Lewis Bowen (Texas A&M) Entropy Theory for Sofic Group Actions 30 / 48

Some strange phenomena f is not monotone under factor maps. Lewis Bowen (Texas A&M) Entropy Theory for Sofic Group Actions 31 / 48

Some strange phenomena f is not monotone under factor maps. f is not well defined if the system does not have a finite entropy generating observable. Lewis Bowen (Texas A&M) Entropy Theory for Sofic Group Actions 31 / 48

Some strange phenomena f is not monotone under factor maps. f is not well defined if the system does not have a finite entropy generating observable. If µ = t µ 1 + ( 1 − t ) µ 2 where µ 1 and µ 2 are invariant and mutually singular then f ( µ, φ ) = tf ( µ 1 , φ ) + ( 1 − t ) f ( µ 2 , φ ) − ( r − 1 ) H ( t ) where H ( t ) = − t log ( t ) − ( 1 − t ) log ( 1 − t ) . Lewis Bowen (Texas A&M) Entropy Theory for Sofic Group Actions 31 / 48

Some strange phenomena f is not monotone under factor maps. f is not well defined if the system does not have a finite entropy generating observable. If µ = t µ 1 + ( 1 − t ) µ 2 where µ 1 and µ 2 are invariant and mutually singular then f ( µ, φ ) = tf ( µ 1 , φ ) + ( 1 − t ) f ( µ 2 , φ ) − ( r − 1 ) H ( t ) where H ( t ) = − t log ( t ) − ( 1 − t ) log ( 1 − t ) . f can take negative values. Lewis Bowen (Texas A&M) Entropy Theory for Sofic Group Actions 31 / 48