Homoclinic Points of Algebraic Z D -actions
Homoclinic Points of Algebraic Z D -actions
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发表时间：
1996
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通讯作者：
D. Lind;K. Schmidt
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作者：
D. Lind;K. Schmidt
设α是Zd通过紧交换群X的连续自同构的作用. X中的点x称为对α同宿的，如果当n → ∞时α nx → 0X。研究了X的子群α的同宿点集<$α（X）.如果α是可扩张的，则<$α（X）至多是可数的。我们的主要结果是：如果α是可扩张的，则（1）<$α（x）是非平凡的当且仅当α是正熵的，（2）<$α（X）是非平凡的且在X中稠密的当且仅当α是完全正熵的.在许多重要的情况下，<$α（X）是由一个基本同宿点生成的，这个基本同宿点可以用傅里叶分析显式计算。同宿点的扩张行动必须衰减到零指数快速，我们用这个来建立强规范的性质，这样的行动。这提供了一个广泛的类的例子Z d-行动Ruelle的热力学形式主义适用。本文最后用一系列的例子突出了扩张性在我们的主要结果中的关键作用。
Let α be an action of Z d by continuous automorphisms of a compact abelian group X. A point x in X is called homoclinic for α if α n x → 0X as n → ∞. We study the set ∆α(X) of homoclinic points for α, which is a subgroup of X. If α is expansive then ∆α(X) is at most countable. Our main results are that if α is expansive, then (1) ∆α(x) is nontrivial if and only if α has positive entropy and (2) ∆α(X) is nontrivial and dense in X if and only if α has completely positive entropy. In many important cases ∆α(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 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.