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Automorphisms of solenoids and \(p\)-adic entropy. (English) Zbl 0634.22005

A solenoid is a finite-dimensional, connected, compact abelian group. We show that a full solenoid is locally the product of a Euclidean component and \(p\)-adic components for each rational prime \(p\). An automorphism of a solenoid preserves these components, and its topological entropy is shown to be the sum of the Euclidean and \(p\)-adic contributions. The \(p\)-adic entropy of the corresponding rational matrix is computed using its \(p\)-adic eigenvalues, and this is used to recover Yuzvinskii’s calculation of entropy for solenoidal automorphisms. The proofs apply Bowen’s investigation of entropy for uniformly continuous transformations to linear maps over the ad‘ele ring of the rationals.

MSC:

22D45 Automorphism groups of locally compact groups
22C05 Compact groups
37P20 Dynamical systems over non-Archimedean local ground fields
28D20 Entropy and other invariants
22D40 Ergodic theory on groups
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References:

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