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July  2017, 37(7): 3701-3719. doi: 10.3934/dcds.2017157

Hermodynamic formalism and k-bonacci substitutions

Aix-Marseille University, I2M, UMR 7373, 13 453 Marseille, France

Received  July 2016 Revised  February 2017 Published  April 2017

We study k-bonacci substitutions through the point of view of thermodynamic formalism. For each substitution we define a renormalization operator associated to it and examine its iterates over potentials in a certain class. We also study the pressure function associated to potentials in this class and prove the existence of a freezing phase transition which is realized by the only ergodic measure on the subshift associated to the substitution.

Citation: Jordan Emme. Hermodynamic formalism and k-bonacci substitutions. Discrete & Continuous Dynamical Systems - A, 2017, 37 (7) : 3701-3719. doi: 10.3934/dcds.2017157
References:
[1]

P. Arnoux and G. Rauzy, Représentation géométrique de suites de complexité 2n + 1, Bull. Soc. Math. France, 119 (1991), 199-215.   Google Scholar

[2]

A. BaravieraR. Leplaideur and A. Lopes, The potential point of view for renormalization, Stoch. Dyn., 12 (2012), 1250005, 34pp.  doi: 10.1142/S0219493712500050.  Google Scholar

[3]

N. Bédaride, P. Hubert and R. Leplaideur, Thermodynamic formalism and substitutions, Preprint, arXiv: 1511.03322. Google Scholar

[4]

R. Bowen, Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms, revised edition, Springer-Verlag, Berlin, 2008.  Google Scholar

[5]

H. Bruin and R. Leplaideur, Renormalization, thermodynamic formalism and quasi-crystals in subshifts, Comm. Math. Phys., 321 (2013), 209-247.  doi: 10.1007/s00220-012-1651-4.  Google Scholar

[6]

H. Bruin and R. Leplaideur, Renormalization, freezing phase transitions and Fibonacci quasicrystals, Ann. Sci. éc. Norm. Supér., 48 (2015), 739-763.   Google Scholar

[7]

J. Cassaigne, Complexité et facteurs spéciaux, Bull. Belg. Math. Soc. Simon Stevin, 4 (1997), 67-88.   Google Scholar

[8]

D. Coronel and J. Rivera-Letelier, Low-temperature phase transitions in the quadratic family, Adv. Math., 248 (2013), 453-494.  doi: 10.1016/j.aim.2013.08.008.  Google Scholar

[9]

F. Durand, Linearly recurrent subshifts have a finite number of non-periodic subshift factors, Ergodic Theory Dynam. Systems, 20 (2000), 1061-1078.  doi: 10.1017/S0143385700000584.  Google Scholar

[10]

B. Mossé, Puissances de mots et reconnaissabilité des points fixes d'une substitution, Theoret. Comput. Sci., 99 (1992), 327-334.  doi: 10.1016/0304-3975(92)90357-L.  Google Scholar

[11]

M. Queffélec, Substitution Dynamical Systems -Spectral Analysis, Lecture Notes in Mathematics, Springer Berlin Heidelberg, (2010).  doi: 10.1007/978-3-642-11212-6.  Google Scholar

[12]

G. Rauzy, Nombres algébriques et substitutions, Bull. Soc. Math. France, 110 (1982), 147-178.   Google Scholar

[13] D. Ruelle, Thermodynamic Formalism. second edition, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 2004.  doi: 10.1017/CBO9780511617546.  Google Scholar
[14]

O. Sarig, Phase transitions for countable Markov shifts, Comm. Math. Phys., 217 (2001), 555-577.  doi: 10.1007/s002200100367.  Google Scholar

show all references

References:
[1]

P. Arnoux and G. Rauzy, Représentation géométrique de suites de complexité 2n + 1, Bull. Soc. Math. France, 119 (1991), 199-215.   Google Scholar

[2]

A. BaravieraR. Leplaideur and A. Lopes, The potential point of view for renormalization, Stoch. Dyn., 12 (2012), 1250005, 34pp.  doi: 10.1142/S0219493712500050.  Google Scholar

[3]

N. Bédaride, P. Hubert and R. Leplaideur, Thermodynamic formalism and substitutions, Preprint, arXiv: 1511.03322. Google Scholar

[4]

R. Bowen, Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms, revised edition, Springer-Verlag, Berlin, 2008.  Google Scholar

[5]

H. Bruin and R. Leplaideur, Renormalization, thermodynamic formalism and quasi-crystals in subshifts, Comm. Math. Phys., 321 (2013), 209-247.  doi: 10.1007/s00220-012-1651-4.  Google Scholar

[6]

H. Bruin and R. Leplaideur, Renormalization, freezing phase transitions and Fibonacci quasicrystals, Ann. Sci. éc. Norm. Supér., 48 (2015), 739-763.   Google Scholar

[7]

J. Cassaigne, Complexité et facteurs spéciaux, Bull. Belg. Math. Soc. Simon Stevin, 4 (1997), 67-88.   Google Scholar

[8]

D. Coronel and J. Rivera-Letelier, Low-temperature phase transitions in the quadratic family, Adv. Math., 248 (2013), 453-494.  doi: 10.1016/j.aim.2013.08.008.  Google Scholar

[9]

F. Durand, Linearly recurrent subshifts have a finite number of non-periodic subshift factors, Ergodic Theory Dynam. Systems, 20 (2000), 1061-1078.  doi: 10.1017/S0143385700000584.  Google Scholar

[10]

B. Mossé, Puissances de mots et reconnaissabilité des points fixes d'une substitution, Theoret. Comput. Sci., 99 (1992), 327-334.  doi: 10.1016/0304-3975(92)90357-L.  Google Scholar

[11]

M. Queffélec, Substitution Dynamical Systems -Spectral Analysis, Lecture Notes in Mathematics, Springer Berlin Heidelberg, (2010).  doi: 10.1007/978-3-642-11212-6.  Google Scholar

[12]

G. Rauzy, Nombres algébriques et substitutions, Bull. Soc. Math. France, 110 (1982), 147-178.   Google Scholar

[13] D. Ruelle, Thermodynamic Formalism. second edition, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 2004.  doi: 10.1017/CBO9780511617546.  Google Scholar
[14]

O. Sarig, Phase transitions for countable Markov shifts, Comm. Math. Phys., 217 (2001), 555-577.  doi: 10.1007/s002200100367.  Google Scholar

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