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) September 17th, 2018 - Wroc� law (Poland)

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–GB] 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) If B is a φ -invariant subgroup of A, then h alg ( φ ) = h alg ( φ ↾ B ) + h alg ( φ A / B ) , where φ A / B : A / B → A / B is induced by φ . [Weiss for ent , Peters, Dikranjan–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 Abelian case Example Let p a prime, A = � Z Z ( p ) and σ : A → A , ( x n ) n ∈ Z �→ ( x n − 1 ) n ∈ Z the right Bernoulli shift. Then h alg ( σ ) = ent ( σ ) = log p . (Here β = σ − 1 is the left Bernoulli shift and h alg ( β ) = h alg ( σ ).) � Z Z ( p ) = � Note that � � Z ( p ) = Z ( p ), Z Z ( p ) and σ = β : � Z Z ( p ) → � Z Z ( p ). Hence, h alg ( σ ) = h top ( � � σ ) = log p . Example Let k > 1 be an integer and consider µ k : Z → Z , x �→ kx . Then h alg ( µ k ) = log k . Note that � Z = T and � µ k = µ k : T → T .

Algebraic entropy for amenable semigroup actions Algebraic entropy for N -actions Abelian case Let f ( x ) = sx n + a n − 1 x n − 1 + . . . + a 0 ∈ Z [ x ] be a primitive polynomial. The Mahler measure of f is � m ( f ) = log s + log | λ i | , | λ i | > 1 where λ i are the roots of f in C . Theorem (Algebraic Yuzvinski Formula) Let n > 0 , φ : Q n → Q n an endomorphism and f φ ( x ) = sx n + a n − 1 x n − 1 + . . . + a 0 ∈ Z [ x ] the characteristic polynomial of φ . Then h alg ( φ ) = m ( f φ ) .

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 , i.e., a net ( F i ) i ∈ I in P f ( S ) such that, for every s ∈ S , | F i s \ F i | lim = 0 . | F i | i ∈ 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 Lemma and Fekete Lemma] Theorem Let S be a cancellative semigroup which is right-amenable (respectively, left-amenable). For every f ∈ L ( S ) (respectively, 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 (respectively, 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 � X by continuous maps. Consider a left action S 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] 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 � A by endomorphisms. consider a left action S For X ∈ P f ( A ) and F ∈ P f ( S ), let � T F ( α, X ) = α ( s )( X ) ∈ P f ( A ) . 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] 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 � A by endomorphisms. consider a left action S 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 � B and S � B / A are induced by α . where S

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 � K by continuous endomorphisms. consider a left action S γ � γ induces a right action � � S , defined by K γ ( s ) = � γ ( s ) : � K → � � K for every s ∈ S ; � γ is the dual action of γ . γ op 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 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 � K by continuous endomorphisms. consider a left action S 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 � L and S � K / L are induced by γ . where S 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 α � A . Let G be an amenable group, A an abelian group, G α ↾ H � A . For H ≤ G consider H If [ G : H ] = k ∈ N , then h alg ( α ↾ H ) = k · h alg ( α ). In particular, h alg ( α ↾ H ) and h alg ( α ) are simultaneously 0. If H is normal, then h alg ( α ) ≤ h alg ( α ↾ H ). Conjecture α � A. For Let G be an amenable group, A an abelian group, G every H ≤ G, h alg ( α ) ≤ h alg ( α ↾ H ) .

Algebraic entropy for amenable semigroup actions Algebraic entropy for amenable semigroups actions Restriction actions Theorem If H is normal and G / H is infinite, h alg ( α ↾ H ) < ∞ implies h alg ( α ) = 0 . Corollary α � A. Let G and A be infinite abelian groups and G If g ∈ G \ { 0 } is such that G / � g � is infinite and h alg ( α ( g )) < ∞ , then h alg ( α ) = 0 . α Hence, for actions Z d � A with d > 1, if h alg ( α ( g )) < ∞ for some g ∈ Z d , g � = 0, then h alg ( α ) = 0; [Eberlein for h top , Conze for h µ ] � Q n has h alg ( α ) = 0. α every action Z d (Compare with the case d = 1, i.e., the Algebraic Yuzvinski Formula.)

Algebraic entropy for amenable semigroup actions Shifts Let G be an amenable group and A an abelian group. Consider the action σ G , A � A G G defined, for every g ∈ G , by σ G , A ( g )( f )( x ) = f ( g − 1 x ) for every f ∈ A G and x ∈ G . In other words, for every ( a x ) x ∈ G ∈ A G , σ G , A (( a x ) x ∈ G ) = ( a g − 1 x ) x ∈ G . If G = Z , then σ Z , A (1) = σ is the right Bernoulli shift, that is, σ (( a n ) n ∈ Z ) = ( a n − 1 ) n ∈ Z .

Algebraic entropy for amenable semigroup actions Shifts Let G be an amenable group and A an abelian group. Consider the action β G , A � A G G defined, for every g ∈ G , by β G , A ( g )( f )( x ) = f ( xg ) for every f ∈ A G and x ∈ G . In other words, for every ( a x ) x ∈ G ∈ A G , β G , A (( a x ) x ∈ G ) = ( a xg ) x ∈ G . If G = Z , then β Z , A (1) = β is the left Bernoulli shift, that is, β (( a n ) n ∈ Z ) = ( a n +1 ) n ∈ Z .