The Ruelle operator for symmetric β –shifts
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Artur O. Lopes
Universidade Federal do Rio Grande do Sul. Instituto de Matemática
Victor Vargas
Universidade Federal do Rio Grande do Sul. Instituto de Matemática
Paraules clau
β -expansions, equilibrium states, Gibbs states, Ruelle operator, symmetric β-shifts
Article Details
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Lopes, Artur O.; Vargas, Victor. «The Ruelle operator for symmetric β –shifts». Publicacions Matemàtiques, 2020, vol.VOL 64, núm. 2, p. 661-80, http://raco.cat/index.php/PublicacionsMatematiques/article/view/371210.
Referències
D. Aguiar, L. Cioletti, and R. Ruviaro, A variational principle for the speci_c entropy for symbolic systems with uncountable alphabets, Math. Nachr.291(17-18) (2018), 2506-2515. DOI: 10.1002/mana.201700229.
R. Alcaraz Barrera, Topological and ergodic properties of symmetric subshifts,Discrete Contin. Dyn. Syst. 34(11) (2014), 4459-4486. DOI: 10.3934/dcds.2014.34.4459.
R. Alcaraz Barrera, S. Baker, and D. Kong, Entropy, topological transitivity,and dimensional properties of unique q-expansions, Trans. Amer. Math.Soc. 371(5) (2019), 3209-3258. DOI: 10.1090/tran/7370.
S. Baker and W. Steiner, On the regularity of the generalised golden ratiofunction, Bull. Lond. Math. Soc. 49(1) (2017), 58-70. DOI: 10.1112/blms.12002.
A. Baraviera, R. Leplaideur, and A. Lopes, \Ergodic Optimization, ZeroTemperature Limits and the Max-Plus Algebra", 29o Col_oquio Brasileiro de Matem_atica, Publica_c~oes Matem_aticas do IMPA, Instituto Nacional de Matem_aticaPura e Aplicada (IMPA), Rio de Janeiro, 2013.
R. Bissacot and E. Garibaldi, Weak KAM methods and ergodic optimal problemsfor countable Markov shifts, Bull. Braz. Math. Soc. (N.S.) 41(3) (2010),321-338. DOI: 10.1007/s00574-010-0014-z.
R. Bissacot and R. dos Santos Freire, Jr., On the existence of maximizingmeasures for irreducible countable Markov shifts: a dynamical proof, ErgodicTheory Dynam. Systems 34(4) (2014), 1103-1115. DOI: 10.1017/etds.2012.194.
L. Cioletti and E. A. Silva, Spectral properties of the Ruelle operator onthe Walters class over compact spaces, Nonlinearity 29(8) (2016), 2253-2278.DOI: 10.1088/0951-7715/29/8/2253.
G. Contreras, A. O. Lopes, and Ph. Thieullen, Lyapunov minimizing measuresfor expanding maps of the circle, Ergodic Theory Dynam. Systems 21(5)(2001), 1379-1409. DOI: 10.1017/S0143385701001663.
P. Erd}os, M. Horv_ath, and I. Jo_o, On the uniqueness of the expansions1 =Pq????ni , Acta Math. Hungar. 58(3-4) (1991), 333-342. DOI: 10.1007/BF01903963.
P. Erd}os, I. Jo_o, and V. Komornik, Characterization of the unique expansions1 =P1i=1 q????ni and related problems, Bull. Soc. Math. France 118(3) (1990),377-390. DOI: 10.24033/bsmf.2151.
R. Freire and V. Vargas, Equilibrium states and zero temperature limiton topologically transitive countable Markov shifts, Trans. Amer. Math. Soc.370(12) (2018), 8451-8465. DOI: 10.1090/tran/7291.
E. Garibaldi, \Ergodic Optimization in the Expanding Case'. Concepts, Toolsand Applications", SpringerBriefs in Mathematics, Springer, Cham, 2017.DOI: 10.1007/978-3-319-66643-3.
P. Glendinning and N. Sidorov, Unique representations of real numbers innon-integer bases, Math. Res. Lett. 8(4) (2001), 535-543. DOI: 10.4310/MRL.2001.v8.n4.a12.
O. Jenkinson, R. D. Mauldin, and M. Urba_nski, Zero temperature limits ofGibbs-equilibrium states for countable alphabet subshifts of _nite type, J. Stat.Phys. 119(3-4) (2005), 765-776. DOI: 10.1007/s10955-005-3035-z.
V. Komornik, D. Kong, and W. Li, Hausdor_ dimension of univoque sets andDevil's staircase, Adv. Math. 305 (2017), 165-196. DOI: 10.1016/j.aim.2016.03.047.
R. Leplaideur, A dynamical proof for the convergence of Gibbs measuresat temperature zero, Nonlinearity 18(6) (2005), 2847-2880. DOI: 10.1088/0951-7715/18/6/023.
D. Lind and B. Marcus, \An Introduction to Symbolic Dynamics and Coding",Cambridge University Press, Cambridge, 1995. DOI: 10.1017/CBO9780511626302.
A. O. Lopes, J. K. Mengue, J. Mohr, and R. R. Souza, Entropy and variationalprinciple for one-dimensional lattice systems with a general a priori probability:positive and zero temperature, Ergodic Theory Dynam. Systems 35(6)(2015), 1925-1961. DOI: 10.1017/etds.2014.15.
R. D. Mauldin and M. Urba_nski, Gibbs states on the symbolic space over anin_nite alphabet, Israel J. Math. 125 (2001), 93-130. DOI: 10.1007/BF02773377.
I. D. Morris, Entropy for zero-temperature limits of Gibbs-equilibrium statesfor countable-alphabet subshifts of _nite type, J. Stat. Phys. 126(2) (2007),315-324. DOI: 10.1007/s10955-006-9215-7.
W. Parry, On the _-expansions of real numbers, Acta Math. Acad. Sci. Hungar.11 (1960), 401-416. DOI: 10.1007/BF02020954.
W. Parry and M. Pollicott, Zeta functions and the periodic orbit structureof hyperbolic dynamics, Ast_erisque 187-188 (1990), 268 pp.
A. R_enyi, Representations for real numbers and their ergodic properties, ActaMath. Acad. Sci. Hungar. 8 (1957), 477-493. DOI: 10.1007/BF02020331.
D. Ruelle, Statistical mechanics of a one-dimensional lattice gas, Comm. Math.Phys. 9(4) (1968), 267-278.
O. M. Sarig, Thermodynamic formalism for countable Markov shifts, ErgodicTheory Dynam. Systems 19(6) (1999), 1565-1593. DOI: 10.1017/S0143385799146820.
K. Thomsen, On the structure of beta shifts, in: \Algebraic and TopologicalDynamics", Contemp. Math. 385, Amer. Math. Soc., Providence, RI, 2005,pp. 321-332. DOI: 10.1090/conm/385/07204.
P. Walters, Equilibrium states for _-transformations and related transformations,Math. Z. 159(1) (1978), 65-88. DOI: 10.1007/BF01174569.
R. Alcaraz Barrera, Topological and ergodic properties of symmetric subshifts,Discrete Contin. Dyn. Syst. 34(11) (2014), 4459-4486. DOI: 10.3934/dcds.2014.34.4459.
R. Alcaraz Barrera, S. Baker, and D. Kong, Entropy, topological transitivity,and dimensional properties of unique q-expansions, Trans. Amer. Math.Soc. 371(5) (2019), 3209-3258. DOI: 10.1090/tran/7370.
S. Baker and W. Steiner, On the regularity of the generalised golden ratiofunction, Bull. Lond. Math. Soc. 49(1) (2017), 58-70. DOI: 10.1112/blms.12002.
A. Baraviera, R. Leplaideur, and A. Lopes, \Ergodic Optimization, ZeroTemperature Limits and the Max-Plus Algebra", 29o Col_oquio Brasileiro de Matem_atica, Publica_c~oes Matem_aticas do IMPA, Instituto Nacional de Matem_aticaPura e Aplicada (IMPA), Rio de Janeiro, 2013.
R. Bissacot and E. Garibaldi, Weak KAM methods and ergodic optimal problemsfor countable Markov shifts, Bull. Braz. Math. Soc. (N.S.) 41(3) (2010),321-338. DOI: 10.1007/s00574-010-0014-z.
R. Bissacot and R. dos Santos Freire, Jr., On the existence of maximizingmeasures for irreducible countable Markov shifts: a dynamical proof, ErgodicTheory Dynam. Systems 34(4) (2014), 1103-1115. DOI: 10.1017/etds.2012.194.
L. Cioletti and E. A. Silva, Spectral properties of the Ruelle operator onthe Walters class over compact spaces, Nonlinearity 29(8) (2016), 2253-2278.DOI: 10.1088/0951-7715/29/8/2253.
G. Contreras, A. O. Lopes, and Ph. Thieullen, Lyapunov minimizing measuresfor expanding maps of the circle, Ergodic Theory Dynam. Systems 21(5)(2001), 1379-1409. DOI: 10.1017/S0143385701001663.
P. Erd}os, M. Horv_ath, and I. Jo_o, On the uniqueness of the expansions1 =Pq????ni , Acta Math. Hungar. 58(3-4) (1991), 333-342. DOI: 10.1007/BF01903963.
P. Erd}os, I. Jo_o, and V. Komornik, Characterization of the unique expansions1 =P1i=1 q????ni and related problems, Bull. Soc. Math. France 118(3) (1990),377-390. DOI: 10.24033/bsmf.2151.
R. Freire and V. Vargas, Equilibrium states and zero temperature limiton topologically transitive countable Markov shifts, Trans. Amer. Math. Soc.370(12) (2018), 8451-8465. DOI: 10.1090/tran/7291.
E. Garibaldi, \Ergodic Optimization in the Expanding Case'. Concepts, Toolsand Applications", SpringerBriefs in Mathematics, Springer, Cham, 2017.DOI: 10.1007/978-3-319-66643-3.
P. Glendinning and N. Sidorov, Unique representations of real numbers innon-integer bases, Math. Res. Lett. 8(4) (2001), 535-543. DOI: 10.4310/MRL.2001.v8.n4.a12.
O. Jenkinson, R. D. Mauldin, and M. Urba_nski, Zero temperature limits ofGibbs-equilibrium states for countable alphabet subshifts of _nite type, J. Stat.Phys. 119(3-4) (2005), 765-776. DOI: 10.1007/s10955-005-3035-z.
V. Komornik, D. Kong, and W. Li, Hausdor_ dimension of univoque sets andDevil's staircase, Adv. Math. 305 (2017), 165-196. DOI: 10.1016/j.aim.2016.03.047.
R. Leplaideur, A dynamical proof for the convergence of Gibbs measuresat temperature zero, Nonlinearity 18(6) (2005), 2847-2880. DOI: 10.1088/0951-7715/18/6/023.
D. Lind and B. Marcus, \An Introduction to Symbolic Dynamics and Coding",Cambridge University Press, Cambridge, 1995. DOI: 10.1017/CBO9780511626302.
A. O. Lopes, J. K. Mengue, J. Mohr, and R. R. Souza, Entropy and variationalprinciple for one-dimensional lattice systems with a general a priori probability:positive and zero temperature, Ergodic Theory Dynam. Systems 35(6)(2015), 1925-1961. DOI: 10.1017/etds.2014.15.
R. D. Mauldin and M. Urba_nski, Gibbs states on the symbolic space over anin_nite alphabet, Israel J. Math. 125 (2001), 93-130. DOI: 10.1007/BF02773377.
I. D. Morris, Entropy for zero-temperature limits of Gibbs-equilibrium statesfor countable-alphabet subshifts of _nite type, J. Stat. Phys. 126(2) (2007),315-324. DOI: 10.1007/s10955-006-9215-7.
W. Parry, On the _-expansions of real numbers, Acta Math. Acad. Sci. Hungar.11 (1960), 401-416. DOI: 10.1007/BF02020954.
W. Parry and M. Pollicott, Zeta functions and the periodic orbit structureof hyperbolic dynamics, Ast_erisque 187-188 (1990), 268 pp.
A. R_enyi, Representations for real numbers and their ergodic properties, ActaMath. Acad. Sci. Hungar. 8 (1957), 477-493. DOI: 10.1007/BF02020331.
D. Ruelle, Statistical mechanics of a one-dimensional lattice gas, Comm. Math.Phys. 9(4) (1968), 267-278.
O. M. Sarig, Thermodynamic formalism for countable Markov shifts, ErgodicTheory Dynam. Systems 19(6) (1999), 1565-1593. DOI: 10.1017/S0143385799146820.
K. Thomsen, On the structure of beta shifts, in: \Algebraic and TopologicalDynamics", Contemp. Math. 385, Amer. Math. Soc., Providence, RI, 2005,pp. 321-332. DOI: 10.1090/conm/385/07204.
P. Walters, Equilibrium states for _-transformations and related transformations,Math. Z. 159(1) (1978), 65-88. DOI: 10.1007/BF01174569.