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Hermodynamic formalism and k-bonacci substitutions
Aix-Marseille University, I2M, UMR 7373, 13 453 Marseille, France |
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.
References:
[1] |
P. Arnoux and G. Rauzy,
Représentation géométrique de suites de complexité 2n + 1, Bull. Soc. Math. France, 119 (1991), 199-215.
|
[2] |
A. Baraviera, R. Leplaideur and A. Lopes,
The potential point of view for renormalization, Stoch. Dyn., 12 (2012), 1250005, 34pp.
doi: 10.1142/S0219493712500050. |
[3] |
N. Bédaride, P. Hubert and R. Leplaideur, Thermodynamic formalism and substitutions, Preprint, arXiv: 1511.03322. |
[4] |
R. Bowen, Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms, revised edition, Springer-Verlag, Berlin, 2008. |
[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. |
[6] |
H. Bruin and R. Leplaideur,
Renormalization, freezing phase transitions and Fibonacci quasicrystals, Ann. Sci. éc. Norm. Supér., 48 (2015), 739-763.
|
[7] |
J. Cassaigne,
Complexité et facteurs spéciaux, Bull. Belg. Math. Soc. Simon Stevin, 4 (1997), 67-88.
|
[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. |
[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. |
[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. |
[11] |
M. Queffélec,
Substitution Dynamical Systems -Spectral Analysis, Lecture Notes in Mathematics, Springer Berlin Heidelberg, (2010).
doi: 10.1007/978-3-642-11212-6. |
[12] |
G. Rauzy,
Nombres algébriques et substitutions, Bull. Soc. Math. France, 110 (1982), 147-178.
|
[13] |
D. Ruelle, Thermodynamic Formalism. second edition, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 2004.
doi: 10.1017/CBO9780511617546.![]() ![]() ![]() |
[14] |
O. Sarig,
Phase transitions for countable Markov shifts, Comm. Math. Phys., 217 (2001), 555-577.
doi: 10.1007/s002200100367. |
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.
|
[2] |
A. Baraviera, R. Leplaideur and A. Lopes,
The potential point of view for renormalization, Stoch. Dyn., 12 (2012), 1250005, 34pp.
doi: 10.1142/S0219493712500050. |
[3] |
N. Bédaride, P. Hubert and R. Leplaideur, Thermodynamic formalism and substitutions, Preprint, arXiv: 1511.03322. |
[4] |
R. Bowen, Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms, revised edition, Springer-Verlag, Berlin, 2008. |
[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. |
[6] |
H. Bruin and R. Leplaideur,
Renormalization, freezing phase transitions and Fibonacci quasicrystals, Ann. Sci. éc. Norm. Supér., 48 (2015), 739-763.
|
[7] |
J. Cassaigne,
Complexité et facteurs spéciaux, Bull. Belg. Math. Soc. Simon Stevin, 4 (1997), 67-88.
|
[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. |
[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. |
[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. |
[11] |
M. Queffélec,
Substitution Dynamical Systems -Spectral Analysis, Lecture Notes in Mathematics, Springer Berlin Heidelberg, (2010).
doi: 10.1007/978-3-642-11212-6. |
[12] |
G. Rauzy,
Nombres algébriques et substitutions, Bull. Soc. Math. France, 110 (1982), 147-178.
|
[13] |
D. Ruelle, Thermodynamic Formalism. second edition, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 2004.
doi: 10.1017/CBO9780511617546.![]() ![]() ![]() |
[14] |
O. Sarig,
Phase transitions for countable Markov shifts, Comm. Math. Phys., 217 (2001), 555-577.
doi: 10.1007/s002200100367. |
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