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.

A self similar action ♣G, Xq is an action G ý X ✝ such that, g ☎ ♣xwq ✏ ♣g ☎ xq♣g⑤x ☎ wq for all w P X ✝. for unique g ☎ x P X and g⑤x 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 g⑤v P G such that g ☎ ♣vwq ✏ ♣g ☎ vq♣g⑤v ☎ wq for all w P X ✝. with g ☎ v ✏ ♣g ☎ v1q♣g⑤v1 ☎ v2q ☎ ☎ ☎ ♣g⑤v1⑤v2☎☎☎⑤vk✁1 ☎ vkq and g⑤v ✏ ♣g⑤v1q⑤v2 ☎ ☎ ☎ ⑤vk

Example: The Grigorchuk group G

(Finitely generated by elements of order 2, intermediate growth, amenable but not elementary-amenable). X ✏ tx, y✉; G ý X ✝ has generators a, b, c, d defined recursively: a ☎ ♣xwq ✏ yw a ☎ ♣ywq ✏ xw b ☎ ♣xwq ✏ x♣a ☎ wq b ☎ ♣ywq ✏ y♣c ☎ wq c ☎ ♣xwq ✏ x♣a ☎ wq c ☎ ♣ywq ✏ y♣d ☎ wq d ☎ ♣xwq ✏ xw d ☎ ♣ywq ✏ y♣b ☎ wq

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, Xq is contracting with nucleus N ✏ te, a, b, c, d✉.

Contracting SSAs, nucleus and Moore diagrams

➓ ♣G, Xq is contracting if there is a finite S ⑨ G such that for

every g P G there exists n P N with tg⑤v : v P X ✝, ⑤v⑤ ➙ n✉ ⑨ S.

➓ The nucleus of a contracting ♣G, Xq is the smallest such S:

N :✏ ↕

gPG ✽

↔

n✏0

tg⑤v : v P X ✝, ⑤v⑤ ➙ n✉.

➓ For g P S (S ⑨ G closed under restriction), the Moore

diagram with vertex set S has a directed edge g

♣x, yq

Ý Ý Ý Ý Ñ h ✏ g⑤x for each self similarity relation g ☎ ♣xwq ✏ y♣h ☎ wq.

Moore diagram for the nucleus of the Grigorchuk group

e ♣y,yq ♣x,xq b a ♣x,yq ♣y,xq ♣x,xq c ♣y,yq ♣x,xq d ♣x,xq ♣y,yq ♣y,yq

Figure: put an edge from g to h ✏ g⑤x with label ♣x, g ☎ xq

a ☎ ♣xwq ✏ yw a ☎ ♣ywq ✏ xw b ☎ ♣xwq ✏ x♣a ☎ wq b ☎ ♣ywq ✏ y♣c ☎ wq c ☎ ♣xwq ✏ x♣a ☎ wq c ☎ ♣ywq ✏ y♣d ☎ wq d ☎ ♣xwq ✏ xw d ☎ ♣ywq ✏ y♣b ☎ wq

SSAs from odometers, integer matrices, basilica group, ...

Odometer: Let X ✏ t0, 1, ☎ ☎ ☎ , N ✁ 1✉, G ✏ tgk : k P Z✉ with g :✏ “add 1 modulo N with carry-over to the right” then g⑤i ✏ e for i ➔ N ✁ 1 and g⑤N✁1 ✏ g. Integer Matrix A: Let X :✏ Zn④♣AtqZn for A P Matn♣Zq, with ⑤ det A⑤ → 1. G ✏ Zd acting by ‘addition modulo ♣AtqZn with carry

• ver to the right’ (uses fixed set of representatives for Zn④♣AtqZn).

♣G, Xq is contracting if ⑤λ⑤ → 1 for all eigenvalues of A. Basilica group: Let X ✏ tx, y✉ and recursively define a and b by a ☎ ♣xwq ✏ y♣b ☎ wq a ☎ ♣ywq ✏ xw b ☎ ♣xwq ✏ x♣a ☎ wq b ☎ ♣ywq ✏ yw The basilica group B is the group generated by ta, b✉, it gives a contracting self similar action.

C*(G) bimodule for ♣G, Xq (after V. Nekrashevych)

Take the usual right-Hilbert C ✝♣Gq-module on X, M ✏ à

xPX

C ✝♣Gq M ✏ tm ✏ ♣mxq : mx P C ✝♣Gq✉, with module action ♣mxq ☎ a ✏ ♣mxaq and inner product ①m, n② ✏ ➳

xPX

m✝

xnx.

Then ♣exqy ✏ 1C ✝♣Gqδy,x gives orthonormal basis elements for M, and there is a left action of C ✝♣Gq on M arising from: Ug♣ex ☎ aq ✏ eg☎x ☎ ♣δg⑤xaq

The C*-algebras T ♣G, Xq and O♣G, Xq

The bimodule C*-algebras have natural presentations: T ♣G, Xq :✏ universal C ✝-algebra with generators tSx : x P X✉ and tUg : g P G✉ such that (T1) S✝

y Sx ✏

★ 1 if x =y if x ✘ y S ù T⑤X⑤ (T2) UgUh ✏ Ugh; U✝

g ✏ Ug✁1;

Ue ✏ 1 U ù C ✝♣Gq (T3) UgSx ✏ Sg☎xUg⑤x self-similarity comm. rels. g ☎ ♣xwq ✏ ♣g ☎ xq♣g⑤x ☎ wq O♣G, Xq :✏ quotient of T ♣G, Xq by the extra relation (O) ➦

xPX ˜

Sx ˜ S✝

x ✏ 1

˜ S ù O⑤X⑤

Spanning set and dynamics

For a word v ✏ x1x2 ☎ ☎ ☎ xn, we let Sv :✏ Sx1Sx2 ☎ ☎ ☎ Sxn.

➓

T ♣G, Xq ✏ spantSvUgS✝

w : v, w P X ✝, g P G✉. ➓ If ♣G, Xq is contracting,

O♣G, Xq ✏ spant˜ SvUg ˜ S✝

w : v, w P X ✝, g P N✉ ➓ The dynamics on T ♣G, Xq, and on O♣G, Xq are defined by

σt♣SvUgS✝

wq ✏ et♣⑤v⑤✁⑤w⑤qSvUgS✝ w ➓ Interested in (KMS) equilibrium states of ♣T ♣G, Xq, σq and of

♣O♣G, Xq, σq.

KMS states

➓ Given a continuous action σ : R Ñ Aut♣Aq, there is a dense

*-subalgebra of σ-analytic elements a P A such that t ÞÑ σt♣aq extends to an entire function z ÞÑ σz♣aq.

➓ Definition

The state ϕ of A satisfies the KMS condition at inverse temperature β P r0, ✽q if whenever a and b are analytic for σ, ϕ♣abq ✏ ϕ♣b σiβ♣aqq.

➓ Note: it suffices to verify the above for elements that span a

dense subalgebra, e.g, in our case, the spanning set tSvUgS✝

w✉

Theorem (L. Raeburn Ramagge Whittaker ’13)

• 1. If β P r0, log ⑤X⑤q, there are no KMSβ states for σ;
• 2. if β P ♣log ⑤X⑤, ✽s, for each normalized trace τ on C ✝♣Gq

define ψβ,τ♣SvUgS✝

wq ✏ 0 if v ✘ w, and

ψβ,τ♣SvUgS✝

v q ✏ ♣1 ✁ ⑤X⑤e✁βq ✽

➳

k✏0

e✁β♣k⑤v⑤q✁ ➳

yPX k g☎y✏y

τ♣δg⑤y q ✠ the map τ ÞÑ ψβ,τ is an affine homeomorphism of Choquet simplices onto the KMSβ states of T ♣G, Xq.

• 3. the KMSlog ⑤X⑤ states of T ♣G, Xq arise from KMS states of

O♣G, Xq; and there is at least this one: ψlog ⑤X⑤♣SvUgS✝

wq ✏

★ ⑤X⑤✁⑤v⑤cg if v ✏ w

• therwise.

If ♣G, Xq is contractible, this is the only one.

There is a KMSlog ⑤X⑤ state of O♣G, Xq given by ψlog ⑤X⑤♣˜ SvUg ˜ S✝

wq ✏

★ ⑤X⑤✁⑤v⑤cg if v ✏ w

• therwise.

If ♣G, Xq is contractible, this is the only one. Hence limβ×log ⑤X⑤ ψβ,τ ✏ ψlog ⑤X⑤ for every τ. What is cg?

The asymptotic proportion of points fixed by g P G

Let τ ✏ usual trace on C ✝♣Gq, i.e. τ♣δgq ✏ 0 unless g ✏ e; then the following limit exists as β × log ⑤X⑤, ψβ,τ♣Ugq ✏ ♣1 ✁ ⑤X⑤e✁βq

✽

➳

k✏0

e✁βk✁ ➳

yPX k g☎y✏y

τ♣δg⑤y q ✠ Ý Ñ cg For each n P N and g P G define F n

g :✏ tv P X n : g ☎ v ✏ v and g⑤v ✏ e✉.

Clearly ➳

yPX k g☎y✏y

τ♣δg⑤y q ✏ ⑤F k

g ⑤ and it turns out that

⑤F k

g ⑤

⑤X k⑤ Õ cg P r0, 1q.

The asymptotic proportion of g-invariant sets.

In the contractive case the same limit is obtained starting from any normalized trace on C ✝♣Gq. For instance, if we use the trace τ1 defined as the integrated version of the trivial representation, τ1♣Ugq ✏ 1 for every g P G, we are led to use the measure of g-invariant sets at level k. So instead of ⑤F k

g ⑤ we need to compute the cardinality of the set

G k

g :✏ tw P X k : g ☎ w ✏ w✉.

This yields the same limit: limkÑ✽

⑤G k

g ⑤

⑤X k⑤ ✏ cg, and again, it suffices

to compute it for g P N. Next, in Mike Whittaker’s talk, we’ll see how to compute ⑤F k

g ⑤ using Moore diagrams.