## On dynamical systems with the specification property

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- by Karl Sigmund PDF
- Trans. Amer. Math. Soc.
**190**(1974), 285-299 Request permission

## 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

*T*invariant 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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*Some systems with unique equilibrium states*(to appear).

*Statistical mechanics on a compact set with*${Z^\nu }$

*action satisfying expansiveness and specification*(to appear).

## Additional Information

- © Copyright 1974 American Mathematical Society
- 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