Entropy and mixing for amenable group actions.

By D. Rudolph, Barak Weiss. 2000

95 citations recorded by the source corpus.

For i a countable amenable group consider those actions of i as measurepreserving transformations of a standard probability space, written asfT∞g∞2i acting on (X;F;„). We sayfT∞g∞2i has completely positive entropy (or simply cpe for short) if for any flnite and nontrivial partition P of X the entropy h(T;P) is not zero. Our goal is to demonstrate what is well known for actions of and even d , that actions of completely positive entropy have very strong mixing properties. Let Si be a list of flnite subsets of i. We say the Si spread if any particular ∞6 id belongs to at most flnitely many of the sets SiS i1 i . Theorem 0.1. For fT∞g∞2i an action of i of completely positive entropy and P any flnite partition, for any sequence of flnite sets Siµ i which spread we have

Subfields: Computational Theory and Mathematics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/9ecd62731eb93d51455cf3b1ab5f7fa3709bb12f

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In Computational Theory and Mathematics

1. Xiangdong Ye
2. Wen Huang
3. Guohua Zhang
4. Hanfeng Li
5. O. N. Ageev
6. Lewis Bowen
7. A. Dooley
8. Eli Glasner

In Geometry and Topology

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2. Hanfeng Li
3. Wen Huang
4. Guohua Zhang
5. David Kerr
6. A. Dooley
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8. Lewis Bowen

In Mathematical Physics

1. Xiangdong Ye
2. Hanfeng Li
3. Wen Huang
4. Guohua Zhang
5. A. Dooley
6. David Kerr
7. Eli Glasner
8. Lewis Bowen

Citations in the corpus

• Compact Group Automorphisms, Addition Formulas and Fuglede-Kadison Determinants (2010, methodology): “The proof for each of these addition formulas employs both topological and measure-theoretical tools, using generalization of the various fibre and conditional entropies studied in [19], and the addition formula for Γ-extensions in [12, 79] which in turn depends on Rudolph and Weiss’s orbit equivale”
• Local Entropy Theory of a Random Dynamical System (2011, background): “However, a turning point was the pioneering paper of Ornstein and Weiss [59] which laid the foundations of the theory of amenable group actions.”
• Local entropy theory for a countable discrete amenable group action (2010, methodology): “The following terminology and results are due to Ornstein and Weiss [35] (see also [41, 44]).”
• Sofic entropy and amenable groups (2010, background): “h that ρ(γ,·) is A-measurable for all γ∈ Γ then the orbit change from T to Sis said to be A-measurable. The smallest such σ-algebra is called the orbit change σ-algebra. The next theorem is proven in [RW00]. 31 Theorem 6.1. Suppose Tis an essentially free ergodic action of a countable discrete amena”
• The homogeneous spectrum problem in ergodic theory (2005, methodology): “2. Let T be a free action of Gn. Since the group Gn is amenable, it follows that, in principle, we can apply the Rokhlin lemma (see [ RW ]).”