An uncountable family of group automorphisms, and a typical member

Thomas Ward

doi:10.1112/S0024609397003330

An uncountable family of group automorphisms, and a typical member

Thomas Ward

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Abstract

We describe an uncountable family of compact group automorphisms with entropy log2. Each member of the family has a distinct dynamical zeta function, and the members are parametrised by a probability space. A positive proportion of the members have positive upper growth rate of periodic points, and almost all of them have an irrational dynamical zeta function. If infinitely many Mersenne numbers have a bounded number of prime divisors, then a typical member of the family has upper growth rate of periodic points equal to log2, and lower growth rate equal to zero.

Original language	English
Pages (from-to)	577-584
Number of pages	8
Journal	Bulletin of the London Mathematical Society
Volume	29
Issue number	5
DOIs	https://doi.org/10.1112/S0024609397003330
Publication status	Published - 1997

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title = "An uncountable family of group automorphisms, and a typical member",

abstract = "We describe an uncountable family of compact group automorphisms with entropy log2. Each member of the family has a distinct dynamical zeta function, and the members are parametrised by a probability space. A positive proportion of the members have positive upper growth rate of periodic points, and almost all of them have an irrational dynamical zeta function. If infinitely many Mersenne numbers have a bounded number of prime divisors, then a typical member of the family has upper growth rate of periodic points equal to log2, and lower growth rate equal to zero.",

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AB - We describe an uncountable family of compact group automorphisms with entropy log2. Each member of the family has a distinct dynamical zeta function, and the members are parametrised by a probability space. A positive proportion of the members have positive upper growth rate of periodic points, and almost all of them have an irrational dynamical zeta function. If infinitely many Mersenne numbers have a bounded number of prime divisors, then a typical member of the family has upper growth rate of periodic points equal to log2, and lower growth rate equal to zero.

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DO - 10.1112/S0024609397003330

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