TY - JOUR

T1 - Universal adic approximation, invariant measures and scaled entropy

AU - Vershik, A. M.

AU - Zatitskii, P. B.

PY - 2017/1/1

Y1 - 2017/1/1

N2 - We define an infinite graded graph of ordered pairs and a canonical action of the group ℤ (the adic action) and of the infinite sum of groups of order two D = Σ∞1 ℤ/2ℤ on the path space of the graph. It is proved that these actions are universal for both groups in the following sense: every ergodic action of these groups with invariant measure and binomial generator, multiplied by a special action (the 'odometer'), is metrically isomorphic to the canonical adic action on the path space of the graph with a central measure. We consider a series of related problems.

AB - We define an infinite graded graph of ordered pairs and a canonical action of the group ℤ (the adic action) and of the infinite sum of groups of order two D = Σ∞1 ℤ/2ℤ on the path space of the graph. It is proved that these actions are universal for both groups in the following sense: every ergodic action of these groups with invariant measure and binomial generator, multiplied by a special action (the 'odometer'), is metrically isomorphic to the canonical adic action on the path space of the graph with a central measure. We consider a series of related problems.

KW - Adic transformation

KW - Graph of ordered pairs

KW - Scaled entropy

KW - Universal action

UR - http://www.scopus.com/inward/record.url?scp=85029726975&partnerID=8YFLogxK

U2 - 10.1070/IM8610

DO - 10.1070/IM8610

M3 - Article

AN - SCOPUS:85029726975

VL - 81

SP - 734

EP - 770

JO - Izvestiya Mathematics

JF - Izvestiya Mathematics

SN - 1064-5632

IS - 4

ER -