Abstract
We show that ergodic automorphisms of compact abelian groups have the property that for every nonempty open setU, the measure of the set first returning toU aftern iterates decays exponentially inn. This follows from a result about aperiodic automorphisms of countable abelian groups, whose proof employsp-adic analysis.
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M. Keane and M. Smorodinsky,Bernoulli schemes of the same entropy are finitarily isomorphic, Ann. of Math. (2),109 (1979), 397–406.
M. Keane and M. Smorodinsky,The finitary isomorphism theorem for Markov shifts, Bull. Am. Math. Soc. (N. S.)1 (1979), 436–438.
Neal Koblitz,p-adic Numbers, p-adic Analysis, and Zeta Functions, Springer, New York, 1977.
L. Kronecker,Zwei Sätze über Gleichungen mit ganzzahligen Coefficienten, J. Reine Angew. Math.53 (1857), 173–175.
D. A. Lind,The structure of skew products with ergodic group automorphisms, Isr. J. Math.28 (1977), 205–248.
D. A. LindSplit skew products, a related functional equation, and specification, Isr. J. Math.30 (1978), 236–254.
D. A. Lind,Dynamical properties of quasihyperbolic toral automorphisms, to appear.
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Partially supported by NSF Grant MCS 7704915.
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Lind, D.A. Ergodic group automorphisms are exponentially recurrent. Israel J. Math. 41, 313–320 (1982). https://doi.org/10.1007/BF02760537
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DOI: https://doi.org/10.1007/BF02760537