American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Metastable energy strata in numerical discretizations of weakly nonlinear wave equations
  • DCDS Home
  • This Issue
  • Next Article
    Classical solvability of the multidimensional free boundary problem for the thin film equation with quadratic mobility in the case of partial wetting
August  2017, 37(7): 3701-3719. doi: 10.3934/dcds.2017157

Hermodynamic formalism and k-bonacci substitutions

Jordan Emme

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.

Keywords: Thermodynamic formalism, word combinatorics, substitutions, pressure function, symbolic dynamics.
Mathematics Subject Classification: Primary: 37D35, 37B10; Secondary: 37C30.
Citation: Jordan Emme. Hermodynamic formalism and k-bonacci substitutions. Discrete & Continuous Dynamical Systems, 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. Baraviera, R. 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. Baraviera, R. 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

[1]

Eugen Mihailescu. Applications of thermodynamic formalism in complex dynamics on $\mathbb{P}^2$. Discrete & Continuous Dynamical Systems, 2001, 7 (4) : 821-836. doi: 10.3934/dcds.2001.7.821
[2]

A. Alamo, J. M. Sanz-Serna. Word combinatorics for stochastic differential equations: Splitting integrators. Communications on Pure & Applied Analysis, 2019, 18 (4) : 2163-2195. doi: 10.3934/cpaa.2019097
[3]

Vaughn Climenhaga. A note on two approaches to the thermodynamic formalism. Discrete & Continuous Dynamical Systems, 2010, 27 (3) : 995-1005. doi: 10.3934/dcds.2010.27.995
[4]

Michael Jakobson, Lucia D. Simonelli. Countable Markov partitions suitable for thermodynamic formalism. Journal of Modern Dynamics, 2018, 13: 199-219. doi: 10.3934/jmd.2018018
[5]

Manfred Denker, Yuri Kifer, Manuel Stadlbauer. Thermodynamic formalism for random countable Markov shifts. Discrete & Continuous Dynamical Systems, 2008, 22 (1&2) : 131-164. doi: 10.3934/dcds.2008.22.131
[6]

Yongluo Cao, De-Jun Feng, Wen Huang. The thermodynamic formalism for sub-additive potentials. Discrete & Continuous Dynamical Systems, 2008, 20 (3) : 639-657. doi: 10.3934/dcds.2008.20.639
[7]

Anna Mummert. The thermodynamic formalism for almost-additive sequences. Discrete & Continuous Dynamical Systems, 2006, 16 (2) : 435-454. doi: 10.3934/dcds.2006.16.435
[8]

Luis Barreira. Nonadditive thermodynamic formalism: Equilibrium and Gibbs measures. Discrete & Continuous Dynamical Systems, 2006, 16 (2) : 279-305. doi: 10.3934/dcds.2006.16.279
[9]

Manfred Denker, Yuri Kifer, Manuel Stadlbauer. Corrigendum to: Thermodynamic formalism for random countable Markov shifts. Discrete & Continuous Dynamical Systems, 2015, 35 (1) : 593-594. doi: 10.3934/dcds.2015.35.593
[10]

Yakov Pesin. On the work of Sarig on countable Markov chains and thermodynamic formalism. Journal of Modern Dynamics, 2014, 8 (1) : 1-14. doi: 10.3934/jmd.2014.8.1
[11]

L. Cioletti, E. Silva, M. Stadlbauer. Thermodynamic formalism for topological Markov chains on standard Borel spaces. Discrete & Continuous Dynamical Systems, 2019, 39 (11) : 6277-6298. doi: 10.3934/dcds.2019274
[12]

Gerhard Keller. Stability index, uncertainty exponent, and thermodynamic formalism for intermingled basins of chaotic attractors. Discrete & Continuous Dynamical Systems - S, 2017, 10 (2) : 313-334. doi: 10.3934/dcdss.2017015
[13]

Renaud Leplaideur. From local to global equilibrium states: Thermodynamic formalism via an inducing scheme. Electronic Research Announcements, 2014, 21: 72-79. doi: 10.3934/era.2014.21.72
[14]

Steven T. Piantadosi. Symbolic dynamics on free groups. Discrete & Continuous Dynamical Systems, 2008, 20 (3) : 725-738. doi: 10.3934/dcds.2008.20.725
[15]

Weinan E, Jianchun Wang. A thermodynamic study of the two-dimensional pressure-driven channel flow. Discrete & Continuous Dynamical Systems, 2016, 36 (8) : 4349-4366. doi: 10.3934/dcds.2016.36.4349
[16]

Jim Wiseman. Symbolic dynamics from signed matrices. Discrete & Continuous Dynamical Systems, 2004, 11 (2&3) : 621-638. doi: 10.3934/dcds.2004.11.621
[17]

George Osipenko, Stephen Campbell. Applied symbolic dynamics: attractors and filtrations. Discrete & Continuous Dynamical Systems, 1999, 5 (1) : 43-60. doi: 10.3934/dcds.1999.5.43
[18]

Michael Hochman. A note on universality in multidimensional symbolic dynamics. Discrete & Continuous Dynamical Systems - S, 2009, 2 (2) : 301-314. doi: 10.3934/dcdss.2009.2.301
[19]

Clark Butler, Kiho Park. Thermodynamic formalism of $ \text{GL}_2(\mathbb{R}) $-cocycles with canonical holonomies. Discrete & Continuous Dynamical Systems, 2021, 41 (5) : 2141-2166. doi: 10.3934/dcds.2020356
[20]

De-Jun Feng, Antti Käenmäki. Equilibrium states of the pressure function for products of matrices. Discrete & Continuous Dynamical Systems, 2011, 30 (3) : 699-708. doi: 10.3934/dcds.2011.30.699

2020 Impact Factor: 1.392

Tools

Metrics

  • PDF downloads (41)
  • HTML views (53)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences | Privacy Policy