# American Institute of Mathematical Sciences

December  2015, 20(10): 3565-3579. doi: 10.3934/dcdsb.2015.20.3565

## Projective distance and $g$-measures

 1 Instituto de Física, Universidad Autónoma de San Luis Potosí, Avenida Manuel Nava 6, Zona Universitaria, 78290 San Luis Potosí, Mexico, Mexico

Received  January 2015 Revised  March 2015 Published  September 2015

We introduce a distance in the space of fully-supported probability measures on one-dimensional symbolic spaces. We compare this distance to the $\bar{d}$-distance and we prove that in general they are not comparable. Our projective distance is inspired on Hilbert's projective metric, and in the framework of $g$-measures, it allows to assess the continuity of the entropy at $g$-measures satisfying uniqueness. It also allows to relate the speed of convergence and the regularity of sequences of locally finite $g$-functions, to the preservation at the limit, of certain ergodic properties for the associate $g$-measures.
Citation: Liliana Trejo-Valencia, Edgardo Ugalde. Projective distance and $g$-measures. Discrete & Continuous Dynamical Systems - B, 2015, 20 (10) : 3565-3579. doi: 10.3934/dcdsb.2015.20.3565
##### References:
 [1] G. Birkhoff, Extensions of Jentzch's theorem,, Transactions of the American Mathematical Society, 85 (1957), 219. Google Scholar [2] M. Bramson and S. Kalikow, Nonuniqueness in $g$-Functions,, Israel Journal of Mathematics, 84 (1993), 153. doi: 10.1007/BF02761697. Google Scholar [3] X. Bressaud, R. Fernández and A. Galves, Speed of $\bard$-convergence for Markov approximations of chains with complete connections. A coupling approach,, Stochastic Processes and Applications, 83 (1999), 127. doi: 10.1016/S0304-4149(99)00025-3. Google Scholar [4] J.-R. Chazottes, E. Floriani and R. Lima, Relative entropy and identification of Gibbs measures in dynamical systems,, Journal of Statistical Physics, 90 (1998), 697. doi: 10.1023/A:1023220802597. Google Scholar [5] J.-R. Chazottes, L. Ramirez and E. Ugalde, Finite type approximations of Gibbs measures on sofic subshifts,, Nonlinearity, 18 (2005), 445. doi: 10.1088/0951-7715/18/1/023. Google Scholar [6] J.-R. Chazottes and E. Ugalde, On the preservation of Gibbsianness under symbol amalgamation,, in Entropy of Hidden Markov Processes and Connections to Dynamical Systems, (2011), 72. Google Scholar [7] Z. Coelho and A. Quas, Criteria for $\bard$-continuity,, Transactions of the American Mathematical Society, 350 (1998), 3257. doi: 10.1090/S0002-9947-98-01923-0. Google Scholar [8] M. Denker, C. Grillenberger and K. Sigmund, Ergodic Theory on Compact Spaces,, Lecture Notes in Mathematics, (1976). Google Scholar [9] F. Dyson, Existence of a phase-transition in a one-dimesional Ising ferromagnet,, Communications in Mathematical Physics, 12 (1969), 91. doi: 10.1007/BF01645907. Google Scholar [10] P. Ferrero and B. Schmitt, Théorème de Ruelle-Perron-Frobenius et Métriques Projectives,, 1979., (). Google Scholar [11] J. Fröhlich and T. Spencer, The phase transition in the one-dimensional Ising model with $1/r^2$ interaction energy,, Communications in Mathematical Physics, 84 (1982), 87. doi: 10.1007/BF01208373. Google Scholar [12] D. Hilbert, Ueber die Gerade Linie als körzeste Verbindung zweier Punkte,, Mathematische Annalen, 46 (1885), 91. Google Scholar [13] P. Hulse, An example of non-unique $g$-measures,, Ergodic Theory and Dynamical Systems, 26 (2006), 439. doi: 10.1017/S0143385705000489. Google Scholar [14] M. Keane, Strongly Mixing $g$-Measures,, Inventiones Mathematicae, 16 (1972), 309. doi: 10.1007/BF01425715. Google Scholar [15] G. Keller, Equilibrium States in Ergodic Theory,, London Mathematical Society, (1998). Google Scholar [16] F. Ledrappier, Principe variationnel et systèmes dynamiques symboliques,, Zeitschrift für Wahrscheinlichkeitstheorie und verwandte Gebiet, 30 (1974), 185. doi: 10.1007/BF00533471. Google Scholar [17] C. Liverani, Decay of correlations,, Annals of Mathematics, 142 (1995), 239. doi: 10.2307/2118636. Google Scholar [18] C. Liverani, Decay of correlations for piecewise expanding maps,, Journal of Statistical Physics, 78 (1995), 1111. doi: 10.1007/BF02183704. Google Scholar [19] C. Liverani, B. Saussol and S. Vaienti, Conformal measure and decay of correlation for covering weighted systems,, Ergodic Theory and Dynamical Systems, 18 (1998), 1399. doi: 10.1017/S0143385798118023. Google Scholar [20] C. Maldonado and R. Salgado-García, Markov approximations of Gibbs measures for long-range interactions on 1D lattices,, Journal of Statistical Mechanics: Theory and Experiment, 2013 (2013). Google Scholar [21] K. Marton, Bounding $\bard$-distance by informational divergence: A method to prove measure concentration,, Annals of Probability, 24 (1996), 857. doi: 10.1214/aop/1039639365. Google Scholar [22] K. Marton, Measure concentration for a class of random processes,, Probability Theory and Related Fields, 110 (1998), 427. doi: 10.1007/s004400050154. Google Scholar [23] V. Maume-Deschamps, Correlation decay for Markov maps on a countable state space,, Ergodic Theory and Dynamical Systems, 21 (2001), 165. doi: 10.1017/S0143385701001110. Google Scholar [24] V. Maume-Deschamps, Projective metric and mixing properties on towers,, Transactions of the American Mathematical Society, 353 (2001), 3371. doi: 10.1090/S0002-9947-01-02786-6. Google Scholar [25] O. Onicescu and G. Mihoc, Sur les Chaînes de variables statistiques,, Bulletin de Sciences Mathématiques, 59 (1935), 174. Google Scholar [26] D. S. Ornstein, An application of ergodic theory to probability theory,, The Annals of Probability, 1 (1973), 43. doi: 10.1214/aop/1176997024. Google Scholar [27] R. Salgado-García and E. Ugalde, Exact scaling in the expansion-modification system,, Journal of Statistical Physics, 153 (2013), 842. doi: 10.1007/s10955-013-0866-x. Google Scholar [28] E. Seneta, Non-negative matrices an Markov Chains,, $2^{nd}$ edition, (1973). Google Scholar [29] P. Shields, Ergodic Theory of Discrete Sample Paths,, Graduate Studies in Mathematics, (1996). Google Scholar [30] P. Walters, Ruelle's operator theorem and $g$-measures,, Transactions of the American Mathematical Society, 214 (1975), 375. Google Scholar

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##### References:
 [1] G. Birkhoff, Extensions of Jentzch's theorem,, Transactions of the American Mathematical Society, 85 (1957), 219. Google Scholar [2] M. Bramson and S. Kalikow, Nonuniqueness in $g$-Functions,, Israel Journal of Mathematics, 84 (1993), 153. doi: 10.1007/BF02761697. Google Scholar [3] X. Bressaud, R. Fernández and A. Galves, Speed of $\bard$-convergence for Markov approximations of chains with complete connections. A coupling approach,, Stochastic Processes and Applications, 83 (1999), 127. doi: 10.1016/S0304-4149(99)00025-3. Google Scholar [4] J.-R. Chazottes, E. Floriani and R. Lima, Relative entropy and identification of Gibbs measures in dynamical systems,, Journal of Statistical Physics, 90 (1998), 697. doi: 10.1023/A:1023220802597. Google Scholar [5] J.-R. Chazottes, L. Ramirez and E. Ugalde, Finite type approximations of Gibbs measures on sofic subshifts,, Nonlinearity, 18 (2005), 445. doi: 10.1088/0951-7715/18/1/023. Google Scholar [6] J.-R. Chazottes and E. Ugalde, On the preservation of Gibbsianness under symbol amalgamation,, in Entropy of Hidden Markov Processes and Connections to Dynamical Systems, (2011), 72. Google Scholar [7] Z. Coelho and A. Quas, Criteria for $\bard$-continuity,, Transactions of the American Mathematical Society, 350 (1998), 3257. doi: 10.1090/S0002-9947-98-01923-0. Google Scholar [8] M. Denker, C. Grillenberger and K. Sigmund, Ergodic Theory on Compact Spaces,, Lecture Notes in Mathematics, (1976). Google Scholar [9] F. Dyson, Existence of a phase-transition in a one-dimesional Ising ferromagnet,, Communications in Mathematical Physics, 12 (1969), 91. doi: 10.1007/BF01645907. Google Scholar [10] P. Ferrero and B. Schmitt, Théorème de Ruelle-Perron-Frobenius et Métriques Projectives,, 1979., (). Google Scholar [11] J. Fröhlich and T. Spencer, The phase transition in the one-dimensional Ising model with $1/r^2$ interaction energy,, Communications in Mathematical Physics, 84 (1982), 87. doi: 10.1007/BF01208373. Google Scholar [12] D. Hilbert, Ueber die Gerade Linie als körzeste Verbindung zweier Punkte,, Mathematische Annalen, 46 (1885), 91. Google Scholar [13] P. Hulse, An example of non-unique $g$-measures,, Ergodic Theory and Dynamical Systems, 26 (2006), 439. doi: 10.1017/S0143385705000489. Google Scholar [14] M. Keane, Strongly Mixing $g$-Measures,, Inventiones Mathematicae, 16 (1972), 309. doi: 10.1007/BF01425715. Google Scholar [15] G. Keller, Equilibrium States in Ergodic Theory,, London Mathematical Society, (1998). Google Scholar [16] F. Ledrappier, Principe variationnel et systèmes dynamiques symboliques,, Zeitschrift für Wahrscheinlichkeitstheorie und verwandte Gebiet, 30 (1974), 185. doi: 10.1007/BF00533471. Google Scholar [17] C. Liverani, Decay of correlations,, Annals of Mathematics, 142 (1995), 239. doi: 10.2307/2118636. Google Scholar [18] C. Liverani, Decay of correlations for piecewise expanding maps,, Journal of Statistical Physics, 78 (1995), 1111. doi: 10.1007/BF02183704. Google Scholar [19] C. Liverani, B. Saussol and S. Vaienti, Conformal measure and decay of correlation for covering weighted systems,, Ergodic Theory and Dynamical Systems, 18 (1998), 1399. doi: 10.1017/S0143385798118023. Google Scholar [20] C. Maldonado and R. Salgado-García, Markov approximations of Gibbs measures for long-range interactions on 1D lattices,, Journal of Statistical Mechanics: Theory and Experiment, 2013 (2013). Google Scholar [21] K. Marton, Bounding $\bard$-distance by informational divergence: A method to prove measure concentration,, Annals of Probability, 24 (1996), 857. doi: 10.1214/aop/1039639365. Google Scholar [22] K. Marton, Measure concentration for a class of random processes,, Probability Theory and Related Fields, 110 (1998), 427. doi: 10.1007/s004400050154. Google Scholar [23] V. Maume-Deschamps, Correlation decay for Markov maps on a countable state space,, Ergodic Theory and Dynamical Systems, 21 (2001), 165. doi: 10.1017/S0143385701001110. Google Scholar [24] V. Maume-Deschamps, Projective metric and mixing properties on towers,, Transactions of the American Mathematical Society, 353 (2001), 3371. doi: 10.1090/S0002-9947-01-02786-6. Google Scholar [25] O. Onicescu and G. Mihoc, Sur les Chaînes de variables statistiques,, Bulletin de Sciences Mathématiques, 59 (1935), 174. Google Scholar [26] D. S. Ornstein, An application of ergodic theory to probability theory,, The Annals of Probability, 1 (1973), 43. doi: 10.1214/aop/1176997024. Google Scholar [27] R. Salgado-García and E. Ugalde, Exact scaling in the expansion-modification system,, Journal of Statistical Physics, 153 (2013), 842. doi: 10.1007/s10955-013-0866-x. Google Scholar [28] E. Seneta, Non-negative matrices an Markov Chains,, $2^{nd}$ edition, (1973). Google Scholar [29] P. Shields, Ergodic Theory of Discrete Sample Paths,, Graduate Studies in Mathematics, (1996). Google Scholar [30] P. Walters, Ruelle's operator theorem and $g$-measures,, Transactions of the American Mathematical Society, 214 (1975), 375. Google Scholar
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