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Periodic points and finite group actions on shifts of finite type

Published online by Cambridge University Press:  19 September 2008

Ulf-Rainer Fiebig
Ulf-Rainer Fiebig
Affiliation:
Institut für Angewandte Mathematik, Universität Heidelberg, Im Neuenheimer Feld 294, 6900 Heidelberg, Germany
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Abstract

Let G be an abstract finite group. For an action α of G on a shift of finite type (SFT) S we introduce the periodic data of α, a computable finite-ordered set of complex polynomials. We show that two actions of G on possibly different SFTs are conjugate on periodic points iff their periodic data coincide. For each subgroup H of G the points fixed by α|H (the restriction of α to H) form a subsystem of S, which is of finite type. Our result shows that the zeta functions of these subsystems determine the conjugacy class (on periodic points) of α up to finitely many possibilities.

The orbit space of a finite skew action on an SFT S, endowed with the homeomorphism induced by S, is shown to have a zeta function equal to the zeta function of an SFT which is a left-closing quotient of S. We show that this zeta function equals the zeta function of S iff the skew action is inert with respect to a certain power of S.

Finally we consider functions of the periodic data as for example gyration numbers.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 13 , Issue 3 , September 1993 , pp. 485 - 514
Copyright
Copyright © Cambridge University Press 1993

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