Homoclinic points of algebraic $\mathbb {Z}^d$-actions

Abstract:

Let $\alpha$ be an action of $\mathbb Z^d$ by continuous automorphisms of a compact abelian group $X$. A point $x$ in $X$ is called homoclinic for $\alpha$ if $\alpha ^{\mathbf n}x\to 0_X$ as $\|\mathbf n\|\to \infty$. We study the set $\Delta _{\alpha }(X)$ of homoclinic points for $\alpha$, which is a subgroup of $X$. If $\alpha$ is expansive, then $\Delta _{\alpha }(X)$ is at most countable. Our main results are that if $\alpha$ is expansive, then (1) $\Delta _{\alpha }(x)$ is nontrivial if and only if $\alpha$ has positive entropy and (2) $\Delta _{\alpha }(X)$ is nontrivial and dense in $X$ if and only if $\alpha$ has completely positive entropy. In many important cases $\Delta _{\alpha }(X)$ is generated by a fundamental homoclinic point which can be computed explicitly using Fourier analysis. Homoclinic points for expansive actions must decay to zero exponentially fast, and we use this to establish strong specification properties for such actions. This provides an extensive class of examples of $\mathbb Z^d$-actions to which Ruelle’s thermodynamic formalism applies. The paper concludes with a series of examples which highlight the crucial role of expansiveness in our main results.