Adjoint entropy vs Topological entropy
Adjoint entropy vs Topological entropy
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伴随熵与拓扑熵
DOI：
10.1016/j.topol.2011.07.032
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发表时间：
2010
期刊：
影响因子：
--
通讯作者：
A. Bruno
中科院分区：
文献类型：
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作者：
A. Bruno
最近，Dikranjan (2010) [6] 引入并研究了阿贝尔群自同态的伴随代数熵。我们将伴随熵的概念推广到拓扑阿贝尔群的连续自同态。事实上，伴随代数熵是使用所有有限索引子群的族来定义的，而我们仅采用所有开放有限索引子群的子族来定义拓扑伴随熵。这使我们能够将拓扑伴随熵与紧阿贝尔群的连续自同态的已知拓扑熵进行比较。特别是，拓扑伴随熵和拓扑熵在完全不连通的紧阿贝尔群的连续自同态上是一致的。此外，我们分别使用庞特里亚金对偶性和预紧对偶性证明了拓扑伴随熵和代数熵之间的两个所谓的桥定理。
Recently the adjoint algebraic entropy of endomorphisms of abelian groups was introduced and studied in Dikranjan (2010) [6]. We generalize the notion of adjoint entropy to continuous endomorphisms of topological abelian groups. Indeed, the adjoint algebraic entropy is defined using the family of all finite-index subgroups, while we take only the subfamily of all open finite-index subgroups to define the topological adjoint entropy. This allows us to compare the topological adjoint entropy with the known topological entropy of continuous endomorphisms of compact abelian groups. In particular, the topological adjoint entropy and the topological entropy coincide on continuous endomorphisms of totally disconnected compact abelian groups. Moreover, we prove two so-called Bridge Theorems between the topological adjoint entropy and the algebraic entropy using respectively the Pontryagin duality and the precompact duality.