Puzzles of Quasi-Finite Type, Zeta Functions and Symbolic Dynamics for Multi-Dimensional Maps
Annales de l'Institut Fourier, Volume 60 (2010) no. 3, pp. 801-852.

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.

Les transformations entropie-dilatantes forment une classe de systèmes dynamiques différentiables généralisant les applications de l’intervalle d’entropie non-nulle et les applications dilatantes. Dans ce travail, on construit une représentation symbolique de ces dynamiques en termes de puzzles (au sens de Yoccoz), évitant ainsi une condition de connexité difficile à satisfaire en dimension supérieure. Ces puzzles sont contrôlés par une « entropie de contrainte » bornée par l’entropie d’hypersurface des transformations précédentes.

L’analyse de ces puzzles repose sur un graphe dénombrable « stablement positif récurrent ». Plus précisément on introduit une « entropie à l’infini » du graphe, contrôlée par l’entropie de contrainte du puzzle, qui permet de généraliser des propriétés classiques des sous-décalages de type fini : multiplicité finie des mesures d’entropie maximale, classification presque topologique, extension méromorphe de fonctions zéta d’Artin-Mazur comptant les points périodiques.

Ces résultats sont enfin appliqués aux puzzles et aux applications entropie-dilatantes « non-dégénérées ».

DOI: 10.5802/aif.2540
Classification: 37B10, 37A35, 37D25, 37C30, 37B40
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
Mot clés : dynamique symbolique, dynamique topologique, théorie ergodique, entropie, mesures d’entropie maximale, points périodiques, fonction zéta d’Artin-Mazur, puzzle, hyperbolicité non-uniforme, transformations entropie-dilatantes, chaînes de Markov topologiques à ensemble d’états dénombrable, récurrence stablement positive, extension méromorphe, conjugaison du point de vue de l’entropie, complexité
Buzzi, Jérôme 1

1 Université Paris-Sud Laboratoire de Mathématique d’Orsay Bât 425 91405 Orsay cedex (France)
@article{AIF_2010__60_3_801_0,
     author = {Buzzi, J\'er\^ome},
     title = {Puzzles of {Quasi-Finite} {Type,} {Zeta} {Functions} and {Symbolic} {Dynamics} for {Multi-Dimensional} {Maps}},
     journal = {Annales de l'Institut Fourier},
     pages = {801--852},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {60},
     number = {3},
     year = {2010},
     doi = {10.5802/aif.2540},
     zbl = {1207.37009},
     mrnumber = {2680817},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/aif.2540/}
}
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
SP  - 801
EP  - 852
VL  - 60
IS  - 3
PB  - Association des Annales de l’institut Fourier
UR  - http://www.numdam.org/articles/10.5802/aif.2540/
DO  - 10.5802/aif.2540
LA  - en
ID  - AIF_2010__60_3_801_0
ER  - 
%0 Journal Article
%A Buzzi, Jérôme
%T Puzzles of Quasi-Finite Type, Zeta Functions and Symbolic Dynamics for Multi-Dimensional Maps
%J Annales de l'Institut Fourier
%D 2010
%P 801-852
%V 60
%N 3
%I Association des Annales de l’institut Fourier
%U http://www.numdam.org/articles/10.5802/aif.2540/
%R 10.5802/aif.2540
%G en
%F AIF_2010__60_3_801_0
Buzzi, Jérôme. Puzzles of Quasi-Finite Type, Zeta Functions and Symbolic Dynamics for Multi-Dimensional Maps. Annales de l'Institut Fourier, Volume 60 (2010) no. 3, pp. 801-852. doi : 10.5802/aif.2540. http://www.numdam.org/articles/10.5802/aif.2540/

[1] Berthé, V. Sequences of low complexity: automatic and Sturmian sequences, Topics in symbolic dynamics and applications (Temuco, 1997) (1-34, London Math. Soc. Lecture Note Ser.), Volume 279, Cambridge Univ. Press, Cambridge (2000) | MR | Zbl

[2] Block, L.; Guckenheimer, J.; Misiurewicz, M.; Young, L.-S. Periodic points and topological entropy of one dimensional maps, Global Theory of Dynamical Systems (Lecture Notes in Math.), Volume 819, Springer, Berlin, 1980, pp. 18-34 | MR | Zbl

[3] Bowen, R. Topological entropy for noncompact sets, Trans. A.M.S., Volume 184 (1975), pp. 125-136 | DOI | MR | Zbl

[4] Boyle, M.; Buzzi, J.; Gomez, R. Almost isomorphism of countable state Markov shifts, Journal fur die reine und angewandte Mathematik, Volume 592 (2006), pp. 23-47 | DOI | MR | Zbl

[5] Branner, B.; Hubbard, J. H. The iteration of cubic polynomials, Part II: Patterns and parapatterns, Acta Math., Volume 169 (1992), pp. 229-325 | DOI | MR | Zbl

[6] Buzzi, J. Intrinsic ergodicity of smooth interval maps, Israel J. Math., Volume 100 (1997), pp. 125-161 | DOI | MR | Zbl

[7] Buzzi, J. Ergodicité intrinsèque de produits fibrés d’applications chaotiques unidimensionelles, Bull. Soc. Math. France, Volume 126 (1998) no. 1, pp. 51-77 | Numdam | MR | Zbl

[8] Buzzi, J. Markov extensions for multi-dimensional dynamical systems, Israel J. Math., Volume 112 (1999), pp. 357-380 | DOI | MR | Zbl

[9] Buzzi, J. On entropy-expanding maps (2000) (Technical report)

[10] Buzzi, J. The coding of non-uniformly expanding maps with an application to endomorphisms of CP k , Ergodic Th. and Dynam. Syst., Volume 23 (2003), pp. 1015 - 1024 | DOI | MR | Zbl

[11] Buzzi, J. Subshifts of quasi-finite type, Invent. Math., Volume 159 (2005), pp. 369-406 | DOI | MR

[12] Buzzi, J.; Ruette, S. Large entropy implies existence of a maximal entropy measure for interval maps, Discrete Contin. Dyn. Syst., Volume 14 (2006) no. 4, pp. 673-688 | DOI | MR | Zbl

[13] de Melo, W; van Strien, S. One-dimensional dynamics, Ergebnisse der Mathematik und ihrer Grenzgebiete (3), 25, Springer-Verlag, Berlin, 1993 | MR | Zbl

[14] Fiebig, D.; Fiebig, U.-R.; Yuri, M. Pressure and equilibrium states for countable state Markov shifts, Israel J. Math., Volume 131 (2002), pp. 221-257 | DOI | MR | Zbl

[15] Gurevič, B. M. Topological entropy of a countable Markov chain, Dokl. Akad. Nauk SSSR, Volume 187 (1969), pp. 715-718 English: Soviet Math. Dokl. 10 (1969), 911–915 | MR | Zbl

[16] Gurevič, B. M. Shift entropy and Markov measures in the space of paths of a countable graph, Dokl. Akad. Nauk SSSR, Volume 192 (1970), pp. 963-965 English: Soviet Math. Dokl. 11 (1970), 744–747 | MR | Zbl

[17] Gurevič, B. M. Stably recurrent nonnegative matrices, Uspekhi Mat. Nauk, Volume 51 (1996) no. 3(309), pp. 195-196 English: Russian Math. Surveys 51 (1996), no. 3, 551–552 | MR | Zbl

[18] Gurevič, B. M.; Savchenko, S. Thermodynamic formalism for symbolic Markov chains with a countable number of states, Uspekhi Mat. Nauk, Volume 53 (1998), pp. 3-106 English: Russian Math. Surveys 53 (1998), no. 2, 245–344 | MR | Zbl

[19] Gurevič, B. M.; Zargaryan, A. S. Conditions for the existence of a maximal measure for a countable symbolic Markov chain, Vestnik Moskov. Univ. Ser. I Mat. Mekh., Volume 103 (1988) no. 5, pp. 14-18 English: Moscow Univ. Math. Bull. 43 (1988), no. 5, 18–2. | MR | Zbl

[20] Hofbauer, F. On intrinsic ergodicity of piecewise monotonic transformations with positive entropy, Israel J. Math., Volume 34 (1979) no. 3, pp. 213-237 (1980) | DOI | MR | Zbl

[21] Hofbauer, F.; Keller, G. Zeta-functions and transfer-operators for piecewise linear transformations, J. Reine Angew. Math., Volume 352 (1984), pp. 100-113 | DOI | MR | Zbl

[22] Ito, Sh.; Murata, H.; Totoki, H. Remarks on the isomorphism theorem for weak Bernoulli transformations in the general case, Publ. Res. Inst. Math. Sci., Volume 7 (1971/1972), pp. 541-580 | DOI | MR | Zbl

[23] Kaloshin, V. Generic diffeomorphisms with superexponential growth of number of periodic orbits, Comm. Math. Phys., Volume 211 (2000) no. 1, pp. 253-271 | DOI | MR | Zbl

[24] Katok, A. Lyapunov exponents, entropy and periodic orbits for diffeomorphisms, Inst. Hautes Etudes Sci. Publ. Math. (1980) no. 51, pp. 137-173 | DOI | Numdam | MR | Zbl

[25] Kitchens, B. P. Symbolic dynamics. One-sided, two-sided and countable state Markov shifts, Universitext, Springer, Berlin, 1998 | MR | Zbl

[26] Lind, D.; Marcus, B. An introduction to symbolic dynamics and coding, Cambridge University Press, Cambridge, 1995 | MR | Zbl

[27] Mauldin, R. D.; Urbański, M. Graph directed Markov systems. Geometry and dynamics of limit sets, Cambridge Tracts in Mathematics, 148, Cambridge University Press, Cambridge, 2003 | MR | Zbl

[28] McMullen, C. T. Complex dynamics and renormalization, Annals of Mathematics Studies, 135, Princeton University Press, Princeton, NJ, 1994 | MR | Zbl

[29] Milnor, J.; Thurston, W. On iterated maps of the interval, Dynamical Systems (Lecture Notes in Mathematics), Volume 1342, Springer, 1988, pp. 465-564 | MR | Zbl

[30] Misiurewicz, M. Topological conditional entropy, Studia Math., Volume 55 (1976) no. 2, pp. 175-200 | MR | Zbl

[31] Pacifico, M. J.; Vieitez, J. Entropy-expansiveness and domination (2006) (D029 See http://www.preprint.impa.br)

[32] Remmert, R.; Kay, L. Classical Topics in Complex Function Theory, Graduate Texts in Mathematics, Springer, 1998 | MR | Zbl

[33] Ruette, S. Mixing C r maps of the interval without maximal measure, Israel J. Math., Volume 127 (2002), pp. 253-277 | DOI | MR

[34] Ruette, S. On the Vere-Jones classification and existence of maximal measures for countable topological Markov chains, Pacific J. Math., Volume 209 (2003) no. 2, pp. 366-380 | DOI | MR | Zbl

[35] Sarig, O. Thermodynamic formalism for countable Markov shifts, Ergodic Theory Dynam. Systems, Volume 19 (1999) no. 6, pp. 1565-1593 | DOI | MR | Zbl

[36] Sarig, O. Phase Transitions for Countable Topological Markov Shifts, Commun. Math. Phys., Volume 217 (2001), pp. 555-577 | DOI | MR | Zbl

[37] Sarig, O. Thermodynamic formalism for null recurrent potentials, Israel J. Math., Volume 121 (2001), pp. 285-311 | DOI | MR | Zbl

[38] Viana, M. Multidimensional nonhyperbolic attractors, Inst. Hautes Etudes Sci. Publ. Math., Volume 85 (1997), pp. 63-96 | DOI | Numdam | MR | Zbl

[39] Walters, P. An introduction to ergodic theory, Graduate Texts in Mathematics, 79, Springer-Verlag, New-York Berlin, 1982 | MR | Zbl

Cited by Sources: