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An Uncountable Family of Group Automorphisms, and a Typical Member
Author(s) -
Ward Thomas
Publication year - 1997
Publication title -
bulletin of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 2.396
H-Index - 48
eISSN - 1469-2120
pISSN - 0024-6093
DOI - 10.1112/s0024609397003330
Subject(s) - uncountable set , mathematics , automorphism , irrational number , group (periodic table) , combinatorics , mersenne prime , discrete group , pure mathematics , discrete mathematics , countable set , geometry , chemistry , organic chemistry
We describe an uncountable family of compact group automorphisms with entropy log 2. 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 log 2, and lower growth rate equal to zero. 1991 Mathematics Subject Classification 22D40, 58F20.