Homoclinic groups, IE groups, and expansive algebraic actions

By Nhan-Phu Chung, Hanfeng Li. 2011

90 citations recorded by the source corpus.

We give algebraic characterizations for expansiveness of algebraic actions of countable groups. The notion of $$p$$p-expansiveness is introduced for algebraic actions, and we show that for countable amenable groups, a finitely presented algebraic action is $$1$$1-expansive exactly when it has finite entropy. We also study the local entropy theory for actions of countable amenable groups on compact groups by automorphisms, and show that the IE group determines the Pinsker factor for such actions. For an expansive algebraic action of a polycyclic-by-finite group on $$X$$X, it is shown that the entropy of the action is equal to the entropy of the induced action on the Pontryagin dual of the homoclinic group, the homoclinic group is a dense subgroup of the IE group, the homoclinic group is nontrivial exactly when the action has positive entropy, and the homoclinic group is dense in $$X$$X exactly when the action has completely positive entropy.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/97536c75a4b601961c60385983382ca3c97f238c

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In Geometry and Topology

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

In Mathematical Physics

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

Citations in the corpus

• ENTROPY, DETERMINANTS, AND L 2 -TORSION (2012, methodology): “It was used in [14] first to study entropy properties of algebraic actions.”
• Stable sets and mean Li–Yorke chaos in positive entropy systems (2012, background): “ asymptotic pair (x,y) by limn→±∞d(Tnx,Tny)=0. It is known that positive entropy does not imply the existence of a non-diagonal twosided asymptotic pair, see [21, Example 3.4] by Lind and Schmidt. In [8] Chung and Li showed that if Γis a polycyclic-by-finite group andΓacts on a compact Abelian group ”
• Combinatorial Independence and Sofic Entropy (2012, methodology): “This set was studied in [32, 33, 40] in the case G = Zd and in [9] for more general G.”
• Local Entropy Theory of a Random Dynamical System (2011, background): “ross orbits of sets in dynamical systems was studied systematically by Kerr and Li in [41, 42] for amenable group actions and in [43] for sofic group actions, and has been discussed by Chung and Li in [14] for amenable group actions on compact groups by automorphisms. Our principal aim in this articl”