Puzzles of Quasi-Finite Type, Zeta Functions and Symbolic Dynamics for Multi-Dimensional Maps

Jérôme Buzzi[1]

  • [1] Université Paris-Sud Laboratoire de Mathématique d’Orsay Bât 425 91405 Orsay cedex (France)

Annales de l’institut Fourier (2010)

  • Volume: 60, Issue: 3, page 801-852
  • ISSN: 0373-0956

Abstract

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Entropy-expanding transformations define a class of smooth dynamics generalizing interval maps with positive entropy and expanding maps. In this work, we build a symbolic representation of those dynamics in terms of puzzles (in Yoccoz’s sense), thus avoiding a connectedness condition, hard to satisfy in higher dimensions. Those puzzles are controled by a «constraint entropy» bounded by the hypersurface entropy of the aforementioned transformations.The analysis of those puzzles rests on a «stably positively recurrent» countable graph. More precisely, we introduce an «entropy at infinity» for such graphs, bounded by the constraint entropy of the puzzle. This allows the generalization of classical properties of subshifts of finite type: finite multiplicity of maximal entropy measures, almost topological classification, meromorphic extension of Artin-Mazur zeta functions counting periodic points.These results are finally applied to puzzles and non-degenerate entropy-expanding maps.

How to cite

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Buzzi, Jérôme. "Puzzles of Quasi-Finite Type, Zeta Functions and Symbolic Dynamics for Multi-Dimensional Maps." Annales de l’institut Fourier 60.3 (2010): 801-852. <http://eudml.org/doc/116293>.

@article{Buzzi2010,
abstract = {Entropy-expanding transformations define a class of smooth dynamics generalizing interval maps with positive entropy and expanding maps. In this work, we build a symbolic representation of those dynamics in terms of puzzles (in Yoccoz’s sense), thus avoiding a connectedness condition, hard to satisfy in higher dimensions. Those puzzles are controled by a «constraint entropy» bounded by the hypersurface entropy of the aforementioned transformations.The analysis of those puzzles rests on a «stably positively recurrent» countable graph. More precisely, we introduce an «entropy at infinity» for such graphs, bounded by the constraint entropy of the puzzle. This allows the generalization of classical properties of subshifts of finite type: finite multiplicity of maximal entropy measures, almost topological classification, meromorphic extension of Artin-Mazur zeta functions counting periodic points.These results are finally applied to puzzles and non-degenerate entropy-expanding maps.},
affiliation = {Université Paris-Sud Laboratoire de Mathématique d’Orsay Bât 425 91405 Orsay cedex (France)},
author = {Buzzi, Jérôme},
journal = {Annales de l’institut Fourier},
keywords = {Symbolic dynamics; topological dynamics; ergodic theory; entropy; measures of maximal entropy; periodic points; Artin-Mazur zeta function; puzzle; non-uniform hyperbolicity; entropy-expanding transformations; countable state topological Markov chains; stable positive recurrence; meromorphic extensions; entropy-conjugacy; complexity; symbolic dynamics},
language = {eng},
number = {3},
pages = {801-852},
publisher = {Association des Annales de l’institut Fourier},
title = {Puzzles of Quasi-Finite Type, Zeta Functions and Symbolic Dynamics for Multi-Dimensional Maps},
url = {http://eudml.org/doc/116293},
volume = {60},
year = {2010},
}

TY - JOUR
AU - Buzzi, Jérôme
TI - Puzzles of Quasi-Finite Type, Zeta Functions and Symbolic Dynamics for Multi-Dimensional Maps
JO - Annales de l’institut Fourier
PY - 2010
PB - Association des Annales de l’institut Fourier
VL - 60
IS - 3
SP - 801
EP - 852
AB - Entropy-expanding transformations define a class of smooth dynamics generalizing interval maps with positive entropy and expanding maps. In this work, we build a symbolic representation of those dynamics in terms of puzzles (in Yoccoz’s sense), thus avoiding a connectedness condition, hard to satisfy in higher dimensions. Those puzzles are controled by a «constraint entropy» bounded by the hypersurface entropy of the aforementioned transformations.The analysis of those puzzles rests on a «stably positively recurrent» countable graph. More precisely, we introduce an «entropy at infinity» for such graphs, bounded by the constraint entropy of the puzzle. This allows the generalization of classical properties of subshifts of finite type: finite multiplicity of maximal entropy measures, almost topological classification, meromorphic extension of Artin-Mazur zeta functions counting periodic points.These results are finally applied to puzzles and non-degenerate entropy-expanding maps.
LA - eng
KW - Symbolic dynamics; topological dynamics; ergodic theory; entropy; measures of maximal entropy; periodic points; Artin-Mazur zeta function; puzzle; non-uniform hyperbolicity; entropy-expanding transformations; countable state topological Markov chains; stable positive recurrence; meromorphic extensions; entropy-conjugacy; complexity; symbolic dynamics
UR - http://eudml.org/doc/116293
ER -

References

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