Chacon's Type Ergodic Transformations with Unbounded Arithmetic Spacers

2013/11/18 by V. V. Ryzhikov, Ryzhikov, V. V.
Mathematics · #Analytic Number Theory Research #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals #Mathematics and Applications #math.DS

paper · pdf · doi:10.48550/arxiv.1311.4524

Ergodic theory

openalex publication_date 2013/11/18 · arxiv created 2013/12/02 · arxiv updated 2013/12/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/04

Abstract

The following generalizations of the Chacon map are proposed: instead of classical constant spacer sequence (0,1,0) let a sequence (0,s_j,0) be one with unbounded s_j. (We mention also an analogue of the historical Chacon map with spacer sequences in the form (0,s_j).) This narrow class of rank-one transformations may be abundant source of open questions. All such constructions have partial rigidity, but some other properties could be different. For root sequence, s_j= [√(j)], (or s_j= [lnj]) the corresponding action is rigid, moreover it possesses all polynomials in its weak closure. In the linear case s_j=j we get (as well as for the classical Chacon transformation) the property of minimal self-joinings (MSJ). We present some observations about MSJ, mild mixing, partial mixing, æ-mixing, absence of factors, triviality of centralizer and spectral primality, state several problems, and mention exponential "self-similar" Chacon transformations and flows on infinite measure spaces.

Citations

• Minimal Self-Joinings, Bounded Constructions, and Weak Closure of Ergodic Actions

Related

• On Mixing Rank One Infinite Transformations
• Nearly finite Chacon Transformation
• On conservative sequences and their application to ergodic multiplier problems
• Spectral properties and approximations of joinings for infinite rank-one actions
• Mixing on a class of rank one transformations