Algebraic entropy for amenable semigroup actions Anna Giordano Bruno - PowerPoint PPT Presentation

Algebraic entropy for amenable semigroup actions Algebraic entropy for amenable semigroup actions Anna Giordano Bruno (joint work with Dikran Dikranjan and Antongiulio Fornasiero) Workshop “Entropies and soficity” January 19th, 2018 - Lyon (France)

Algebraic entropy for amenable semigroup actions Algebraic entropy for N -actions Abelian case Let A be an abelian group and φ : A → A an endomorphism; P f ( A ) = { F ⊆ A | F � = ∅ finite } ⊇ F ( A ) = { F ≤ A | F finite } . For F ∈ P f ( A ), n > 0, let T n ( φ, F ) = F + φ ( F ) + . . . + φ n − 1 ( F ). The algebraic entropy of φ with respect to F is log | T n ( φ, F ) | H alg ( φ, F ) = lim . n n →∞ [Adler-Konheim-McAndrew, M.Weiss] The algebraic entropy of φ is ent ( φ ) = sup { H alg ( φ, F ) | F ∈ F ( A ) } . [Peters, Dikranjan] The algebraic entropy of φ is h alg ( φ ) = sup { H alg ( φ, F ) | F ∈ P f ( A ) } . Clearly, ent ( φ ) = ent ( φ ↾ t ( A ) ) = h alg ( φ ↾ t ( A ) ) ≤ h alg ( φ ) .

Algebraic entropy for amenable semigroup actions Algebraic entropy for N -actions Abelian case [Dikranjan-Goldsmith-Salce-Zanardo for ent , D-GB for h alg ] Theorem (Addition Theorem = Yuzvinski’s addition formula) If B is a φ -invariant subgroup of A, then h alg ( φ ) = h ( φ ↾ B ) + h ( φ A / B ) , where φ A / B : A / B → A / B is induced by φ . [Weiss for ent , Peters, D-GB for h alg ] Theorem (Bridge Theorem) Denote � A the Pontryagin dual of A and � φ : � A → � A the dual of φ . Then h alg ( φ ) = h top ( � φ ) . Here h top denotes the topological entropy for continuous selfmaps of compact spaces [Adler-Konheim-McAndrew].

Algebraic entropy for amenable semigroup actions Algebraic entropy for N -actions Non-abelian case Non-abelian case Let G be a group and φ : G → G an endomorphism. Let P f ( G ) = { F ⊆ G | F � = ∅ finite } . For F ∈ P f ( G ), n > 0, let T n ( φ, F ) = F · φ ( F ) · . . . · φ n − 1 ( F ) . The algebraic entropy of φ with respect to F is log | T n ( φ, F ) | H alg ( φ, F ) = lim . n n →∞ [Dikranjan-GB] The algebraic entropy of φ is h alg ( φ ) = sup { H alg ( φ, F ) | F ∈ P f ( G ) } .

Algebraic entropy for amenable semigroup actions Algebraic entropy for N -actions Non-abelian case G = � X � finitely generated group ( X ∈ P f ( G )). For g ∈ G \ { 1 } , ℓ X ( g ) is the length of the shortest word representing g in X ∪ X − 1 , and ℓ X (1) = 0. For n ≥ 0, let B X ( n ) = { g ∈ G | ℓ X ( g ) ≤ n } . The growth function of G wrt X is γ X : N → N , n �→ | B X ( n ) | . log γ X ( n ) The growth rate of G wrt X is λ X = lim n →∞ . n For φ = id G and 1 ∈ X , T n ( id G , X ) = B X ( n ) and H alg ( id G , X ) = λ X . [Milnor Problem, Grigorchuk group, Gromov Theorem] There exists a group of intermediate growth. G has polynomial growth if and only if G is virtually nilpotent.

Algebraic entropy for amenable semigroup actions Algebraic entropy for N -actions Non-abelian case Let G be a group, φ : G → G an endomorphism and X ∈ P f ( G ). The growth rate of φ wrt X is γ φ, X : N + → N + , n �→ | T n ( φ, X ) | . If G = � X � with 1 ∈ X ∈ P f ( G ), then γ X = γ id G , X . φ has polynomial growth if γ φ, X is polynomial ∀ X ∈ P f ( G ); φ has exponential growth if ∃ F ∈ P f ( G ), γ φ, X is exp.; φ has intermediate growth otherwise. φ has exponential growth if and only if h alg ( φ ) > 0 . The Addition Theorem does not hold for h alg : let G = Z ( Z ) ⋊ β Z ; G has exponential growth and so h alg ( id G ) = ∞ ; Z ( Z ) and Z are abelian and hence h alg ( id Z ( Z ) ) = 0 = h alg ( id Z ). Theorem ([GB-Spiga, Dikranjan-GB for abelian groups, Milnor-Wolf in the classical setting]) No endomorphism of a locally virtually soluble group has intermediate growth.

Algebraic entropy for amenable semigroup actions Ornstein-Weiss Lemma for semigroups Let S be a cancellative semigroup. S is right-amenable if and only if S admits a right-Følner net , | F i s \ F i | i.e., a net ( F i ) i ∈ I in P f ( S ) such that lim i ∈ I = 0 ∀ s ∈ S . | F i | (analogously, left-amenable). A map f : P f ( S ) → R is: 1 subadditive if f ( F 1 ∪ F 2 ) ≤ f ( F 1 ) + f ( F 2 ) ∀ F 1 , F 2 ∈ P f ( S ); 2 left-subinvariant if f ( sF ) ≤ f ( F ) ∀ s ∈ S ∀ F ∈ P f ( S ); 3 right-subinvariant if f ( Fs ) ≤ f ( F ) ∀ s ∈ S ∀ F ∈ P f ( S ); 4 unif. bounded on singletons if ∃ M ≥ 0 , f ( { s } ) ≤ M ∀ s ∈ S . Let L ( S ) = { f : P f ( S ) → R | (1) , (2) , (4) hold for f } and R ( S ) = { f : P f ( S ) → R | (1) , (3) , (4) hold for f } .

Algebraic entropy for amenable semigroup actions Ornstein-Weiss Lemma for semigroups [Ceccherini Silberstein-Coornaert-Krieger, generalizing Ornstein-Weiss Theorem] Let S be a cancellative right-amenable (resp., left-amenable) semigroup. For every f ∈ L ( S ) (resp., f ∈ R ( S )) there exists λ ∈ R ≥ 0 such that f ( F i ) H S ( f ) := lim | F i | = λ i ∈ I for every right-Følner (resp., left-Følner) net ( F i ) i ∈ I of S .

Algebraic entropy for amenable semigroup actions Amenable semigroups actions Topological entropy Let S be a cancellative left-amenable semigroup, X a compact space and cov ( X ) the family of all open covers of X . For U ∈ cov ( X ), let N ( U ) = min {|V| | V ⊆ U} . γ Consider a left action S � X by continuous maps. For U ∈ cov ( X ) and F ∈ P f ( S ), let � γ ( s ) − 1 ( U ) ∈ cov ( X ) . U γ, F = s ∈ F f U : P fin ( S ) → R , F �→ log N ( U γ, F ) . Then f U ∈ R ( S ). [Ceccherini-Silberstein-Coornaert-Krieger, gen. Moulin Ollagnier] The topological entropy of γ with respect to U is H top ( γ, U ) = H S ( f U ) . The topological entropy of γ is h top ( γ ) = sup { H top ( γ, U ) | U ∈ cov ( X ) } .

Algebraic entropy for amenable semigroup actions Amenable semigroups actions Algebraic entropy Let S be a cancellative right-amenable semigroup. Let A be an abelian group and α consider a left action S � A by endomorphisms. For X ∈ P f ( A ) and F ∈ P f ( S ), let � α ( s )( X ) ∈ P f ( A ) . T F ( α, X ) = s ∈ F f X : P fin ( S ) → R , F �→ log | T F ( α, X ) | . Then f X ∈ L ( S ). The algebraic entropy of α with respect to X is H alg ( α, X ) = H S ( f X ) . [Fornasiero-GB-Dikranjan, Virili for groups] The algebraic entropy of α is h alg ( α ) = sup { H alg ( α, X ) | X ∈ P f ( A ) } . Moreover, ent ( α ) = sup { H alg ( α, X ) | X ∈ F ( A ) } .

Algebraic entropy for amenable semigroup actions Algebraic entropy for amenable semigroups actions Addition Theorem Let S be a cancellative right-amenable semigroup. Let A be an abelian group and α consider a left action S � A by endomorphisms. Theorem (Addition Theorem) If A is torsion and B is an α -invariant subgroup of A, then h alg ( α ) = h alg ( α B ) + h alg ( α A / B ) , α B / A α B where S � B and S � B / A are induced by α .

Algebraic entropy for amenable semigroup actions Algebraic entropy for amenable semigroups actions Bridge Theorem Let S be a cancellative left-amenable semigroup. Let K be a compact abelian group and γ consider a left action S � K by continuous endomorphisms. γ � γ induces a right action � K � S , defined by γ ( s ) = � γ ( s ) : � K → � � K for every s ∈ S ; � γ is the dual action of γ . � γ γ op the left action S op � � Denote by � K associated to � γ of the cancellative right-amenable semigroup S op . Theorem (Bridge Theorem) If K is totally disconnected (i.e., A is torsion), then γ op ) . h top ( γ ) = h alg ( � [Virili for amenable group actions on locally compact abelian groups]

Algebraic entropy for amenable semigroup actions Algebraic entropy for amenable semigroups actions Bridge Theorem Let S be a cancellative left-amenable semigroup. Let K be a compact abelian group and γ consider a left action S � K by continuous endomorphisms. Corollary (Addition Theorem) If K is totally disconnected and L is a γ -invariant subgroup of K, then h top ( γ ) = h top ( γ L ) + h top ( γ K / L ) , γ K / L γ L where S � L and S � K / L are induced by γ . Known in the case of compact groups for: Z d -actions on compact groups [Lind-Schmidt-Ward]; actions of countable amenable groups on compact metrizable groups [Li].

Algebraic entropy for amenable semigroup actions Algebraic entropy for amenable semigroups actions Restriction actions and quotient actions Restriction and quotient actions α Let G be an amenable group, A an abelian group, G � A . α ↾ H For H ≤ G consider H � A . If [ G : H ] = k ∈ N , then h alg ( α ↾ H ) = k · h alg ( α ). If H is normal, then h alg ( α ) ≤ h alg ( α ↾ H ). α G / N ¯ For N ≤ G normal with N ⊆ ker α , consider G / N � A . � 0 if N is infinite , h alg ( α ) = h alg ( α G / N ) if N is finite . | N | Corollary h alg ( α G / ker α ) If h alg ( α ) > 0 , then ker α is finite and h alg ( α ) = . | ker α | So: reduction to faithful actions.