4,611 research outputs found
Subshifts of quasi-finite type
We introduce subshifts of quasi-finite type as a generalization of the
well-known subshifts of finite type. This generalization is much less rigid and
therefore contains the symbolic dynamics of many non-uniform systems, e.g.,
piecewise monotonic maps of the interval with positive entropy. Yet many
properties remain: existence of finitely many ergodic invariant probabilities
of maximum entropy; lots of periodic points; meromorphic extension of the
Artin-Mazur zeta function.Comment: added examples, more precise estimates on periodic points and
classificatio
Puzzles of Quasi-Finite Type, Zeta Functions and Symbolic Dynamics for Multi-Dimensional Maps
We introduce "puzzles of quasi-finite type" which are the counterparts of our
subshifts of quasi-finite type (Invent. Math. 159 (2005)) in the setting of
combinatorial puzzles as defined in complex dynamics. We are able to analyze
these dynamics defined by entropy conditions rather completely, obtaining a
complete classification with respect to large entropy measures and a
description of their measures with maximum entropy and periodic orbits. These
results can in particular be applied to entropy-expanding maps like
(x,y)-->(1.8-x^2+sy,1.9-y^2+sx) for small s. We prove in particular the
meromorphy of the Artin-Mazur zeta function on a large disk. This follows from
a similar new result about strongly positively recurrent Markov shifts where
the radius of meromorphy is lower bounded by an "entropy at infinity" of the
graph.Comment: accepted by Annales de l'Institut Fourier, final revised versio
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