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  • Ergodic Theory and Dynamical Systems, Volume 8, Issue 3
  • September 1988, pp. 411-419

Automorphisms of solenoids and p-adic entropy*

  • D. A. Lind (a1) and T. Ward (a2)
  • DOI:
  • Published online: 01 September 2008

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 adele ring of the rationals.

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[Be]K. Berg . Convolution of invariant measure, maximal entropy. Math. Systems Theory 3 (1969), 46150.

[B]R. Bowen . Entropy for group endomorphisms and homogeneous spaces. Trans. Amer. Math. Soc. 153 (1971), 401414.

[H]P. R. Halmos . On automorphisms of compact groups. Bull. Amer. Math. Soc. 49 (1943), 619624.

[K]N. Koblitz . p-adic Numbers, p-adic Analysis, and Zeta Functions. Springer: New York, 1977.

[Lan]S. Lang . Fundamentals of Diophantine Geometry. Springer: New York, 1983.

[La]W. Lawton . The structure of compact connected groups which admit an expansive automorphism. Springer Lect. Notes in Math. 318 (1973), 182196.

[L2]D. Lind . Ergodic group automorphisms are exponentially recurrent. Israel J. Math. 41 (1982), 313320.

[P]J. Peters . Entropy on discrete abelian groups. Advances in Math. 33 (1979), 113.

[Wa]P. Walters . An Introduction to Ergodic Theory. Springer: New York, 1982.

[We]A. Weil . Basic Number Theory 3rd ed., Springer: New York, 1974.

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