Journal ArticleDOI
Ergodic automorphisms of the infinite torus are bernoulli
TLDR
In this paper, it was shown that ergodic algebraic automorphisms of the infinite torus are measure isomorphic to Bernoulli shifts, and that the existence of such an automorphism with finite entropy is equivalent to an open problem in algebraic number theory.Abstract:
We show that ergodic algebraic automorphisms of the infinite torus are measure isomorphic to Bernoulli shifts. Using the same techniques, we also show that the existence of such an automorphism with finite entropy is equivalent to an open problem in algebraic number theory.read more
Citations
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Journal ArticleDOI
Mahler measure and entropy for commuting automorphisms of compact groups.
TL;DR: In this article, the joint entropy of commuting automorphisms of a compact metrizable group has been shown to be the sum of the topological entropy and the number of periodic points in a compact group.
Journal ArticleDOI
Speculations concerning the range of Mahler's measure
TL;DR: In this paper, the authors describe Smyth's lecture at the University of British Columbia and present a number of proofs which would not be otherwise accessible as Appendix A. The attentive reader will soon realize the appropriateness of the title.
Journal ArticleDOI
Nonuniformly hyperbolic K-systems are Bernoulli
Nikolai Chernov,C. Haskell +1 more
TL;DR: In this article, it was shown that non-uniformly hyperbolic maps and flows with singularities enjoy the K-property of the Lorentz gas and are Bernoulli properties.
Book ChapterDOI
Number Theory and Polynomials: The Mahler measure of algebraic numbers: a survey
TL;DR: A survey of results for Mahler measure of algebraic numbers, and one-variable polynomials with integer coefficients is presented in this article, where the maximum modulus of the conjugates of an algebraic integer are also discussed.
Journal ArticleDOI
The structure of skew products with ergodic group automorphisms
TL;DR: In this article, it was shown that ergodic automorphisms of compact groups are Bernoulli shifts, and that skew products with such automomorphisms are isomorphic to direct products, and the set of possible values for entropy is one of two alternatives depending on the answer to an open problem in algebraic number theory.
References
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Journal ArticleDOI
Entropy for group endomorphisms and homogeneous spaces
TL;DR: In this article, a topological entropy for affine maps of Lie groups and certain homogeneous spaces is defined and compared with measure theoretic entropy for Haar measure and affine map of compact metrizable groups.
Journal ArticleDOI
Zwei Sätze über Gleichungen mit ganzzahligen Coefficienten.
TL;DR: In this paper, the authors define Wurzeln der Einheit as "the Gleichung in which a, b, c,... lauter imaginäre Gröfsen with dem Modul Eins sein sollen, so setze man, indem man }/
Journal ArticleDOI
On automorphisms of compact groups
TL;DR: In this article, the Pontrjagin duality theory is used to construct a measure-preserving transformation of a character group G. In other words a is a measure preserving transformation of G; the purpose of this note is to investigate a few simple properties of a from the point of view of measure theory.