Quasi-factors and relative entropy for infinite-measure-preserving transformations

2011 ◽  
Vol 185 (1) ◽  
pp. 43-60
Author(s):  
Tom Meyerovitch
Keyword(s):  
Relative Entropy ◽  
Infinite Measure ◽  
Measure Preserving Transformations

Piecewise isometries, uniform distribution and 3log 2−π2/8

2011 ◽  
Vol 32 (6) ◽  
pp. 1862-1888 ◽  
Author(s):  
YITWAH CHEUNG ◽  
AREK GOETZ ◽  
ANTHONY QUAS
Keyword(s):  
Phase Space ◽  
Uniform Distribution ◽  
Periodic Orbits ◽  
Periodic Points ◽  
Full Measure ◽  
Infinite Measure ◽  
Important Family ◽  
Large Numbers ◽  
Piecewise Isometries ◽  
Measure Preserving Transformations

AbstractWe use analytic tools to study a simple family of piecewise isometries of the plane parameterized by an angle. In previous work, we showed the existence of large numbers of periodic points, each surrounded by a ‘periodic island’. We also proved conservativity of the systems as infinite measure-preserving transformations. In experiments it is observed that the periodic islands fill up a large part of the phase space and it has been asked whether the periodic islands form a set of full measure. In this paper we study the periodic islands around an important family of periodic orbits and demonstrate that for all angle parameters that are irrational multiples of π, the islands have asymptotic density in the plane of 3log 2−π2/8≈0.846.

Conjugates of Infinite Measure Preserving Transformations

1988 ◽  
Vol 40 (3) ◽  
pp. 742-749
Author(s):  
S. Alpern ◽  
J. R. Choksi ◽  
V. S. Prasad
Keyword(s):  
Conjugacy Class ◽  
Ergodic Theory ◽  
Lebesgue Space ◽  
Finite Measure ◽  
Infinite Measure ◽  
Ergodic Automorphism ◽  
Measurable Set ◽  
Image Position ◽  
Measure Preserving Transformations

In this paper we consider a question concerning the conjugacy class of an arbitrary ergodic automorphism σ of a sigma finite Lebesgue space (X, , μ) (i.e., a is a ju-preserving bimeasurable bijection of (X, , μ). Specifically we proveTHEOREM 1. Let τ, σ be any pair of ergodic automorphisms of an infinite sigma finite Lebesgue space (X, , μ). Let F be any measurable set such thatThen there is some conjugate σ' of σ such that σ'(x) = τ(x) for μ-almost every x in F.The requirement that F ∪ τF has a complement of infinite measure is, for example, satisfied when F has finite measure, and in that case, the theorem was proved by Choksi and Kakutani ([7], Theorem 6).Conjugacy theorems of this nature have proved to be very useful in proving approximation results in ergodic theory. These conjugacy results all assert the denseness of the conjugacy class of an ergodic (or antiperiodic) automorphism in various topologies and subspaces.

Return- and hitting-time limits for rare events of null-recurrent Markov maps

2015 ◽  
Vol 37 (1) ◽  
pp. 244-276 ◽  
Author(s):  
FRANÇOISE PÈNE ◽  
BENOÎT SAUSSOL ◽  
ROLAND ZWEIMÜLLER
Keyword(s):  
Fixed Points ◽  
Rare Events ◽  
Hitting Time ◽  
Abstract Result ◽  
Limit Distributions ◽  
Interval Maps ◽  
Time Limits ◽  
Infinite Measure ◽  
Asymptotic Type ◽  
Measure Preserving Transformations

We determine limit distributions for return- and hitting-time functions of certain asymptotically rare events for conservative ergodic infinite measure preserving transformations with regularly varying asymptotic type. Our abstract result applies, in particular, to shrinking cylinders around typical points of null-recurrent renewal shifts and infinite measure preserving interval maps with neutral fixed points.

Sign in / Sign up

Export Citation Format

Share Document