scholarly journals A class of ergodic transformations having simple spectrum

1971 ◽  
Vol 27 (2) ◽  
pp. 275-275 ◽  
Author(s):  
J. R. Baxter
Keyword(s):  
Simple Spectrum ◽  
Ergodic Transformations

Ergodic transformations conjugate to their inverses by involutions

1996 ◽  
Vol 16 (1) ◽  
pp. 97-124 ◽  
Author(s):  
Geoffrey R. Goodson ◽  
Andrés del Junco ◽  
Mariusz Lemańczyk ◽  
Daniel J. Rudolph
Keyword(s):  
Discrete Spectrum ◽  
Probability Space ◽  
Closure Property ◽  
Simple Spectrum ◽  
Ergodic Transformations ◽  
Ergodic Automorphism ◽  
Singular Spectrum ◽  
Weak Closure ◽  
Identity Automorphism ◽  
Borel Probability

AbstractLetTbe an ergodic automorphism defined on a standard Borel probability space for whichTandT−1are isomorphic. We investigate the form of the conjugating automorphism. It is well known that ifTis ergodic having a discrete spectrum andSis the conjugation betweenTandT−1, i.e.SsatisfiesTS=ST−1thenS2=Ithe identity automorphism. We show that this result remains true under the weaker assumption thatThas a simple spectrum. IfThas the weak closure property and is isomorphic to its inverse, it is shown that the conjugationSsatisfiesS4=I. Finally, we construct an example to show that the conjugation need not be an involution in this case. The example we construct, in addition to having the weak closure property, is of rank two, rigid and simple for all orders with a singular spectrum of multiplicity equal to two.

scholarly journals ℤd-actions with prescribed topological and ergodic properties

2011 ◽  
Vol 32 (1) ◽  
pp. 191-209 ◽  
Author(s):  
YURI LIMA
Keyword(s):  
Topological Entropy ◽  
Topological Dynamics ◽  
Ergodic Transformations ◽  
Positive Topological Entropy ◽  
Ergodic Properties ◽  
Uniquely Ergodic

AbstractWe extend constructions of Hahn and Katznelson [On the entropy of uniquely ergodic transformations. Trans. Amer. Math. Soc.126 (1967), 335–360] and Pavlov [Some counterexamples in topological dynamics. Ergod. Th. & Dynam. Sys.28 (2008), 1291–1322] to ℤd-actions on symbolic dynamical spaces with prescribed topological and ergodic properties. More specifically, we describe a method to build ℤd-actions which are (totally) minimal, (totally) strictly ergodic and have positive topological entropy.

Infinite Ergodic Transformations

2014 ◽  
pp. 25-39
Author(s):  
Stanley Eigen ◽  
Arshag Hajian ◽  
Yuji Ito ◽  
Vidhu Prasad
Keyword(s):  
Ergodic Transformations

scholarly journals Occupation times of sets of infinite measure for ergodic transformations

2005 ◽  
Vol 25 (4) ◽  
pp. 959-976 ◽  
Author(s):  
JON AARONSON ◽  
MAXIMILIAN THALER ◽  
ROLAND ZWEIMÜLLER
Keyword(s):  
Occupation Times ◽  
Infinite Measure ◽  
Ergodic Transformations

The behavior of Bernoulli shifts relative to their factors

1999 ◽  
Vol 19 (5) ◽  
pp. 1255-1280 ◽  
Author(s):  
CHRISTOPHER HOFFMAN
Keyword(s):  
Bernoulli Shift ◽  
Bernoulli Shifts ◽  
Ergodic Transformations ◽  
Zero Entropy ◽  
Almost All

We present numerous examples of ways that a Bernoulli shift can behave relative to a family of factors. This shows the similarities between the properties which collections of ergodic transformations can have and the behavior of a Bernoulli shift relative to a collection of its factors. For example, we construct a family of factors of a Bernoulli shift which have the same entropy, and any extension of one of these factors has more entropy, yet no two of these factors sit the same. This is the relative analog of Ornstein and Shields uncountable collection of nonisomorphic $K$ transformations of the same entropy. We are able to construct relative analogs of almost all the zero entropy counter-examples constructed by Rudolph (1979), as well as the $K$ counterexamples constructed by Hoffman (1997). This paper provides a solution to a problem posed by Ornstein (1975).

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