Hostname: page-component-76fb5796d-45l2p Total loading time: 0 Render date: 2024-04-28T03:09:25.468Z Has data issue: false hasContentIssue false
  • English
  • Français

Lifting generic points

Published online by Cambridge University Press:  05 February 2024

TOMASZ DOWNAROWICZ Open the ORCID record for TOMASZ DOWNAROWICZ [Opens in a new window]  and
BENJAMIN WEISS Open the ORCID record for BENJAMIN WEISS [Opens in a new window]
TOMASZ DOWNAROWICZ*
Affiliation:
Faculty of Pure and Applied Mathematics, Wrocław University of Technology, Wrocław, Poland
BENJAMIN WEISS
Affiliation:
Einstein Institute of Mathematics, The Hebrew University of Jerusalem, Jerusalem, Israel (e-mail: weiss@math.huji.ac.il)
*
e-mail: downar@pwr.edu.pl
Get access Rights & Permissions [Opens in a new window]

Abstract

Let $(X,T)$ and $(Y,S)$ be two topological dynamical systems, where $(X,T)$ has the weak specification property. Let $\xi $ be an invariant measure on the product system $(X\times Y, T\times S)$ with marginals $\mu $ on X and $\nu $ on Y, with $\mu $ ergodic. Let $y\in Y$ be quasi-generic for $\nu $. Then there exists a point $x\in X$ generic for $\mu $ such that the pair $(x,y)$ is quasi-generic for $\xi $. This is a generalization of a similar theorem by T. Kamae, in which $(X,T)$ and $(Y,S)$ are full shifts on finite alphabets.

Type
Original Article
Information
Ergodic Theory and Dynamical Systems , First View , pp. 1 - 16
Check for updates
Copyright
© The Author(s), 2024. Published by Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Bergelson, V. and Downarowicz, T.. On preservation of normality and determinism under arithmetic operations. In preparation.Google Scholar
Dateyama, M.. The almost weak specification property for ergodic group automorphisms of abelian groups. J. Math. Soc. Japan 42 (1990), 341–351.CrossRefGoogle Scholar
Furstenberg, H.. Strict ergodicity and transformation of the torus. Amer. J. Math. 83 (1961), 573–601.CrossRefGoogle Scholar
Furstenberg, H. and Weiss, B.. On almost $1$ - $1$ extensions. Israel J. Math. 65 (1989), 311–322.CrossRefGoogle Scholar
Kamae, T.. Subsequences of normal sequences. Israel J. Math. 16 (1973), 121–149.CrossRefGoogle Scholar
Kamae, T.. Normal numbers and ergodic theory. Proc. Third Japan--USSR Symp. on Probability Theory (Lecture Notes in Mathematics, 550). Ed. G. Maruyama and J. V. Prokhorov. Springer, Berlin, 1976.Google Scholar
Rauzy, G.. Nombres normaux et processus déterministes. Acta Arith. 29 (1976), 211–225.CrossRefGoogle Scholar
Weiss, B.. Normal sequences as collectives. Proc. Symp. on Topological Dynamics and Ergodic Theory. University of Kentucky, Lexington, KY, 1971, pp. 79–80.Google Scholar