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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

On dynamical systems with the specification property


Author: Karl Sigmund
Journal: Trans. Amer. Math. Soc. 190 (1974), 285-299
MSC: Primary 28A65; Secondary 54H20
DOI: https://doi.org/10.1090/S0002-9947-1974-0352411-X
MathSciNet review: 0352411
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Abstract: A continuous transformation T of a compact metric space X satisfies the specification property if one can approximate distinct pieces of orbits by single periodic orbits with a certain uniformity. There are many examples of such transformations which have recently been studied in ergodic theory and statistical mechanics. This paper investigates the relation between Tinvariant measures and the frequencies of T-orbits. In particular, it is shown that every invariant measure (and even every closed connected subset of such measures) has generic points, but that the set of all generic points is of first category in X. This generalizes number theoretic results concerning decimal expansions and normal numbers.


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DOI: https://doi.org/10.1090/S0002-9947-1974-0352411-X
Keywords: Ergodic measures, generic points, quasiregular points, normal numbers
Article copyright: © Copyright 1974 American Mathematical Society