Hostname: page-component-cd9895bd7-mkpzs Total loading time: 0 Render date: 2024-12-28T01:34:03.520Z Has data issue: false hasContentIssue false
  • English
  • Français

An endomorphism whose square is Bernoulli

Published online by Cambridge University Press:  09 March 2004

CHRISTOPHER HOFFMAN
CHRISTOPHER HOFFMAN
Affiliation:
Department of Mathematics, University of Washington, Seattle, WA 98195, USA (e-mail: hoffman@math.washington.edu)
Get access Rights & Permissions [Opens in a new window]

Abstract

One of the corollaries of Ornstein's isomorphism theorem is that if $(Y,S,\nu)$ is an invertible measure-preserving transformation and $(Y,S^2,\nu)$ is isomorphic to a Bernoulli shift, then $(Y,S,\nu)$ is isomorphic to a Bernoulli shift. In this paper we show that non-invertible transformations do not share this property. We do this by exhibiting a uniformly two-to-one endomorphism $(X,\sigma,\mu)$ which is not isomorphic to the one-sided Bernoulli two shift. However, $(X,T^2,\mu)$ is isomorphic to the one-sided Bernoulli four shift.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 24 , Issue 2 , April 2004 , pp. 477 - 494
Copyright
2004 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.)