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Homoclinic points of algebraic \(\mathbb Z^d\)-actions. (English) Zbl 0940.22004

According to the authors’ terminology an algebraic \(\mathbb Z^d\)-action is an action of \(\mathbb Z^d\) by continuous automorphisms on a compact abelian group. The purpose of the paper under review is to study the homoclinic points of algebraic \(\mathbb Z^d\)-actions. Let \(\alpha\) be an algebraic \(\mathbb Z^d\)-action on the compact abelian group \(X\), and let \(O_X\) be the additive identity of \(X\). A point \(x\in X\) is homoclinic for \(\alpha\) if \(\alpha^{\mathbf n} x\to O_X\) as \(\|{\mathbf n}\|\to\infty\). The set \(\Delta_\alpha(X)\) of all homoclinic points for \(\alpha\) is a subgroup of \(X\) which is called the homoclinic group of \(\alpha\). The two main results of the paper are the following: If \(\alpha\) is an expansive algebraic \(\mathbb Z^d\)-action then (1) \(\Delta_\alpha(X)\) is nontrivial if and only if \(\alpha\) has strictly positive entropy, and (2) \(\Delta_\alpha(X)\) is dense in \(X\) if and only if \(\alpha\) has completely positive entropy. Furthermore, they show that expansive algebraic \(\mathbb Z^d\)-actions with completely positive entropy always satisfy very strong specification, an orbit tracing property studied by Ruelle who showed that there is a thermodynamic formalism for such actions. Finally, the authors describe examples of nonexpansive algebraic \(\mathbb Z^d\)-actions which show that in general there is no relationship between entropy and the size of the homoclinic group.

MSC:

22D40 Ergodic theory on groups
54H20 Topological dynamics (MSC2010)
37D35 Thermodynamic formalism, variational principles, equilibrium states for dynamical systems
13C10 Projective and free modules and ideals in commutative rings
43A75 Harmonic analysis on specific compact groups
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