Equilibrium states for self-similar actions Marcelo Laca Victoria - PowerPoint PPT Presentation

Equilibrium states for self-similar actions Marcelo Laca Victoria COSy, Toronto 28 May 2013 joint work with I. Raeburn, J. Ramagge, and M. Whittaker Equilibrium states on the Cuntz-Pimsner algebras of self-similar actions http://front.math.ucdavis.edu/1301.4722

Self similar group actions: G ✏ a group, X ✏ a finite set X n set of words of length n , X 0 ✏ t ∅ ✉ , X ✝ : ✏ ➈ ✽ n ✏ 0 X n . X ✝ such that, A self similar action ♣ G , X q is an action G ý for all w P X ✝ . g ☎ ♣ xw q ✏ ♣ g ☎ x q♣ g ⑤ x ☎ w q for unique g ☎ x P X and g ⑤ x P G (the restriction of g to x ). We may replace the letter x by an initial word v : for g P G and v P X k there exists a unique g ⑤ v P G such that for all w P X ✝ . g ☎ ♣ vw q ✏ ♣ g ☎ v q♣ g ⑤ v ☎ w q with g ☎ v ✏ ♣ g ☎ v 1 q♣ g ⑤ v 1 ☎ v 2 q ☎ ☎ ☎ ♣ g ⑤ v 1 ⑤ v 2 ☎☎☎ ⑤ v k ✁ 1 ☎ v k q and g ⑤ v ✏ ♣ g ⑤ v 1 q⑤ v 2 ☎ ☎ ☎ ⑤ v k

Example: The Grigorchuk group G (Finitely generated by elements of order 2, intermediate growth, amenable but not elementary-amenable). X ✝ has generators a , b , c , d defined recursively: X ✏ t x , y ✉ ; G ý a ☎ ♣ xw q ✏ yw a ☎ ♣ yw q ✏ xw b ☎ ♣ xw q ✏ x ♣ a ☎ w q b ☎ ♣ yw q ✏ y ♣ c ☎ w q c ☎ ♣ xw q ✏ x ♣ a ☎ w q c ☎ ♣ yw q ✏ y ♣ d ☎ w q d ☎ ♣ xw q ✏ xw d ☎ ♣ yw q ✏ y ♣ b ☎ w q Proposition The generators a, b, c, d of G all have order two, and satisfy cd ✏ b ✏ dc, db ✏ c ✏ bd and bc ✏ d ✏ cb. The self-similar action ♣ G , X q is contracting with nucleus N ✏ t e , a , b , c , d ✉ .

Contracting SSAs, nucleus and Moore diagrams ➓ ♣ G , X q is contracting if there is a finite S ⑨ G such that for every g P G there exists n P N with t g ⑤ v : v P X ✝ , ⑤ v ⑤ ➙ n ✉ ⑨ S . ➓ The nucleus of a contracting ♣ G , X q is the smallest such S : ✽ ↕ ↔ t g ⑤ v : v P X ✝ , ⑤ v ⑤ ➙ n ✉ . N : ✏ g P G n ✏ 0 ➓ For g P S ( S ⑨ G closed under restriction), the Moore diagram with vertex set S has a directed edge ♣ x , y q g Ý Ý Ý Ý Ñ h ✏ g ⑤ x for each self similarity relation g ☎ ♣ xw q ✏ y ♣ h ☎ w q .

Moore diagram for the nucleus of the Grigorchuk group ♣ x , x q a c ♣ x , x q ♣ y , y q ♣ x , y q ♣ y , x q ♣ y , y q b ♣ y , y q ♣ y , y q e d ♣ x , x q ♣ x , x q Figure: put an edge from g to h ✏ g ⑤ x with label ♣ x , g ☎ x q a ☎ ♣ xw q ✏ yw a ☎ ♣ yw q ✏ xw b ☎ ♣ xw q ✏ x ♣ a ☎ w q b ☎ ♣ yw q ✏ y ♣ c ☎ w q c ☎ ♣ xw q ✏ x ♣ a ☎ w q c ☎ ♣ yw q ✏ y ♣ d ☎ w q d ☎ ♣ xw q ✏ xw d ☎ ♣ yw q ✏ y ♣ b ☎ w q

SSAs from odometers, integer matrices, basilica group, ... Odometer: Let X ✏ t 0 , 1 , ☎ ☎ ☎ , N ✁ 1 ✉ , G ✏ t g k : k P Z ✉ with g : ✏ “add 1 modulo N with carry-over to the right” then g ⑤ i ✏ e for i ➔ N ✁ 1 and g ⑤ N ✁ 1 ✏ g . Integer Matrix A : Let X : ✏ Z n ④♣ A t q Z n for A P Mat n ♣ Z q , with ⑤ det A ⑤ → 1. G ✏ Z d acting by ‘addition modulo ♣ A t q Z n with carry over to the right’ (uses fixed set of representatives for Z n ④♣ A t q Z n ). ♣ G , X q is contracting if ⑤ λ ⑤ → 1 for all eigenvalues of A . Basilica group: Let X ✏ t x , y ✉ and recursively define a and b by a ☎ ♣ xw q ✏ y ♣ b ☎ w q a ☎ ♣ yw q ✏ xw b ☎ ♣ xw q ✏ x ♣ a ☎ w q b ☎ ♣ yw q ✏ yw The basilica group B is the group generated by t a , b ✉ , it gives a contracting self similar action.

C*(G) bimodule for ♣ G , X q (after V. Nekrashevych) Take the usual right-Hilbert C ✝ ♣ G q -module on X , C ✝ ♣ G q à M ✏ x P X M ✏ t m ✏ ♣ m x q : m x P C ✝ ♣ G q✉ , with module action ♣ m x q ☎ a ✏ ♣ m x a q and inner product ➳ m ✝ ① m , n ② ✏ x n x . x P X Then ♣ e x q y ✏ 1 C ✝ ♣ G q δ y , x gives orthonormal basis elements for M , and there is a left action of C ✝ ♣ G q on M arising from: U g ♣ e x ☎ a q ✏ e g ☎ x ☎ ♣ δ g ⑤ x a q

The C*-algebras T ♣ G , X q and O ♣ G , X q The bimodule C*-algebras have natural presentations: T ♣ G , X q : ✏ universal C ✝ -algebra with generators t S x : x P X ✉ and t U g : g P G ✉ such that ★ 1 if x =y S ✝ (T1) y S x ✏ S ù T ⑤ X ⑤ 0 if x ✘ y U ✝ C ✝ ♣ G q (T2) U g U h ✏ U gh ; g ✏ U g ✁ 1 ; U e ✏ 1 U ù (T3) U g S x ✏ S g ☎ x U g ⑤ x self-similarity comm. rels. g ☎ ♣ xw q ✏ ♣ g ☎ x q♣ g ⑤ x ☎ w q O ♣ G , X q : ✏ quotient of T ♣ G , X q by the extra relation x P X ˜ S x ˜ ˜ S ✝ ➦ (O) x ✏ 1 S ù O ⑤ X ⑤

Spanning set and dynamics For a word v ✏ x 1 x 2 ☎ ☎ ☎ x n , we let S v : ✏ S x 1 S x 2 ☎ ☎ ☎ S x n . ➓ T ♣ G , X q ✏ span t S v U g S ✝ w : v , w P X ✝ , g P G ✉ . ➓ If ♣ G , X q is contracting, O ♣ G , X q ✏ span t ˜ S v U g ˜ S ✝ w : v , w P X ✝ , g P N ✉ ➓ The dynamics on T ♣ G , X q , and on O ♣ G , X q are defined by σ t ♣ S v U g S ✝ w q ✏ e t ♣⑤ v ⑤✁⑤ w ⑤q S v U g S ✝ w ➓ Interested in (KMS) equilibrium states of ♣ T ♣ G , X q , σ q and of ♣ O ♣ G , X q , σ q .

KMS states ➓ Given a continuous action σ : R Ñ Aut ♣ A q , there is a dense *-subalgebra of σ -analytic elements a P A such that t ÞÑ σ t ♣ a q extends to an entire function z ÞÑ σ z ♣ a q . ➓ Definition The state ϕ of A satisfies the KMS condition at inverse temperature β P r 0 , ✽q if whenever a and b are analytic for σ , ϕ ♣ ab q ✏ ϕ ♣ b σ i β ♣ a qq . ➓ Note: it suffices to verify the above for elements that span a dense subalgebra, e.g, in our case, the spanning set t S v U g S ✝ w ✉

Theorem (L. Raeburn Ramagge Whittaker ’13) 1. If β P r 0 , log ⑤ X ⑤q , there are no KMS β states for σ ; 2. if β P ♣ log ⑤ X ⑤ , ✽s , for each normalized trace τ on C ✝ ♣ G q define ψ β,τ ♣ S v U g S ✝ w q ✏ 0 if v ✘ w, and ✽ e ✁ β ♣ k �⑤ v ⑤q ✁ ➳ ✠ ψ β,τ ♣ S v U g S ✝ v q ✏ ♣ 1 ✁ ⑤ X ⑤ e ✁ β q ➳ τ ♣ δ g ⑤ y q k ✏ 0 y P X k g ☎ y ✏ y the map τ ÞÑ ψ β,τ is an affine homeomorphism of Choquet simplices onto the KMS β states of T ♣ G , X q . 3. the KMS log ⑤ X ⑤ states of T ♣ G , X q arise from KMS states of O ♣ G , X q ; and there is at least this one: ★ ⑤ X ⑤ ✁⑤ v ⑤ c g if v ✏ w ψ log ⑤ X ⑤ ♣ S v U g S ✝ w q ✏ 0 otherwise. If ♣ G , X q is contractible, this is the only one.

There is a KMS log ⑤ X ⑤ state of O ♣ G , X q given by ★ ⑤ X ⑤ ✁⑤ v ⑤ c g if v ✏ w ψ log ⑤ X ⑤ ♣ ˜ S v U g ˜ S ✝ w q ✏ 0 otherwise. If ♣ G , X q is contractible, this is the only one. Hence lim β × log ⑤ X ⑤ ψ β,τ ✏ ψ log ⑤ X ⑤ for every τ . What is c g ?

The asymptotic proportion of points fixed by g P G Let τ ✏ usual trace on C ✝ ♣ G q , i.e. τ ♣ δ g q ✏ 0 unless g ✏ e ; then the following limit exists as β × log ⑤ X ⑤ , ✽ e ✁ β k ✁ ➳ ✠ ➳ ψ β,τ ♣ U g q ✏ ♣ 1 ✁ ⑤ X ⑤ e ✁ β q τ ♣ δ g ⑤ y q Ý Ñ c g k ✏ 0 y P X k g ☎ y ✏ y For each n P N and g P G define g : ✏ t v P X n : g ☎ v ✏ v and g ⑤ v ✏ e ✉ . F n ➳ τ ♣ δ g ⑤ y q ✏ ⑤ F k Clearly g ⑤ and it turns out that y P X k g ☎ y ✏ y ⑤ F k g ⑤ ⑤ X k ⑤ Õ c g P r 0 , 1 q .

The asymptotic proportion of g -invariant sets. In the contractive case the same limit is obtained starting from any normalized trace on C ✝ ♣ G q . For instance, if we use the trace τ 1 defined as the integrated version of the trivial representation, τ 1 ♣ U g q ✏ 1 for every g P G , we are led to use the measure of g -invariant sets at level k . So instead of ⑤ F k g ⑤ we need to compute the cardinality of the set g : ✏ t w P X k : g ☎ w ✏ w ✉ . G k ⑤ G k g ⑤ This yields the same limit: lim k Ñ✽ ⑤ X k ⑤ ✏ c g , and again, it suffices to compute it for g P N . Next, in Mike Whittaker’s talk, we’ll see how to compute ⑤ F k g ⑤ using Moore diagrams.