Projective distance and g-measures

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

Projective distance and $g$-measures

Liliana Trejo-Valencia ^1, and Edgardo Ugalde ^1,

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

Abstract
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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.

Keywords: Markov approximations, $\bar{d}$-distance, $g$-measures, projective distance, entropy convergence..

Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C3.

Citation: Liliana Trejo-Valencia, Edgardo Ugalde. Projective distance and $g$-measures. Discrete and 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), 219-227.
[2]	M. Bramson and S. Kalikow, Nonuniqueness in $g$-Functions, Israel Journal of Mathematics, 84 (1993), 153-160. doi: 10.1007/BF02761697.
[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), 127-138. doi: 10.1016/S0304-4149(99)00025-3.
[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), 697-725. doi: 10.1023/A:1023220802597.
[5]	J.-R. Chazottes, L. Ramirez and E. Ugalde, Finite type approximations of Gibbs measures on sofic subshifts, Nonlinearity, 18 (2005), 445-463. doi: 10.1088/0951-7715/18/1/023.
[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, Cambridge University Press, 2011, 72-97.
[7]	Z. Coelho and A. Quas, Criteria for $\bard$-continuity, Transactions of the American Mathematical Society, 350 (1998), 3257-3268. doi: 10.1090/S0002-9947-98-01923-0.
[8]	M. Denker, C. Grillenberger and K. Sigmund, Ergodic Theory on Compact Spaces, Lecture Notes in Mathematics, 527, Springer-Verlag, 1976.
[9]	F. Dyson, Existence of a phase-transition in a one-dimesional Ising ferromagnet, Communications in Mathematical Physics, 12 (1969), 91-107. doi: 10.1007/BF01645907.
[10]	P. Ferrero and B. Schmitt, Théorème de Ruelle-Perron-Frobenius et Métriques Projectives, 1979.
[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), 87-101. doi: 10.1007/BF01208373.
[12]	D. Hilbert, Ueber die Gerade Linie als körzeste Verbindung zweier Punkte, Mathematische Annalen, 46 (1885), 91-96.
[13]	P. Hulse, An example of non-unique $g$-measures, Ergodic Theory and Dynamical Systems, 26 (2006), 439-445. doi: 10.1017/S0143385705000489.
[14]	M. Keane, Strongly Mixing $g$-Measures, Inventiones Mathematicae, 16 (1972), 309-324. doi: 10.1007/BF01425715.
[15]	G. Keller, Equilibrium States in Ergodic Theory, London Mathematical Society, Student Texts, 42, 1998.
[16]	F. Ledrappier, Principe variationnel et systèmes dynamiques symboliques, Zeitschrift für Wahrscheinlichkeitstheorie und verwandte Gebiet, 30 (1974), 185-202. doi: 10.1007/BF00533471.
[17]	C. Liverani, Decay of correlations, Annals of Mathematics, 142 (1995), 239-301. doi: 10.2307/2118636.
[18]	C. Liverani, Decay of correlations for piecewise expanding maps, Journal of Statistical Physics, 78 (1995), 1111-1129. doi: 10.1007/BF02183704.
[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), 1399-1420. doi: 10.1017/S0143385798118023.
[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), P08012.
[21]	K. Marton, Bounding $\bard$-distance by informational divergence: A method to prove measure concentration, Annals of Probability, 24 (1996), 857-866. doi: 10.1214/aop/1039639365.
[22]	K. Marton, Measure concentration for a class of random processes, Probability Theory and Related Fields, 110 (1998), 427-439. doi: 10.1007/s004400050154.
[23]	V. Maume-Deschamps, Correlation decay for Markov maps on a countable state space, Ergodic Theory and Dynamical Systems, 21 (2001), 165-196. doi: 10.1017/S0143385701001110.
[24]	V. Maume-Deschamps, Projective metric and mixing properties on towers, Transactions of the American Mathematical Society, 353 (2001), 3371-3389. doi: 10.1090/S0002-9947-01-02786-6.
[25]	O. Onicescu and G. Mihoc, Sur les Chaînes de variables statistiques, Bulletin de Sciences Mathématiques, 59 (1935), 174-192.
[26]	D. S. Ornstein, An application of ergodic theory to probability theory, The Annals of Probability, 1 (1973), 43-65. doi: 10.1214/aop/1176997024.
[27]	R. Salgado-García and E. Ugalde, Exact scaling in the expansion-modification system, Journal of Statistical Physics, 153 (2013), 842-863. doi: 10.1007/s10955-013-0866-x.
[28]	E. Seneta, Non-negative matrices an Markov Chains, $2^{nd}$ edition, Springer-Verlag, 1973.
[29]	P. Shields, Ergodic Theory of Discrete Sample Paths, Graduate Studies in Mathematics, 13, American Mathematical Society, 1996.
[30]	P. Walters, Ruelle's operator theorem and $g$-measures, Transactions of the American Mathematical Society, 214 (1975), 375-387.

show all references

References:

[1]	G. Birkhoff, Extensions of Jentzch's theorem, Transactions of the American Mathematical Society, 85 (1957), 219-227.
[2]	M. Bramson and S. Kalikow, Nonuniqueness in $g$-Functions, Israel Journal of Mathematics, 84 (1993), 153-160. doi: 10.1007/BF02761697.
[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), 127-138. doi: 10.1016/S0304-4149(99)00025-3.
[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), 697-725. doi: 10.1023/A:1023220802597.
[5]	J.-R. Chazottes, L. Ramirez and E. Ugalde, Finite type approximations of Gibbs measures on sofic subshifts, Nonlinearity, 18 (2005), 445-463. doi: 10.1088/0951-7715/18/1/023.
[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, Cambridge University Press, 2011, 72-97.
[7]	Z. Coelho and A. Quas, Criteria for $\bard$-continuity, Transactions of the American Mathematical Society, 350 (1998), 3257-3268. doi: 10.1090/S0002-9947-98-01923-0.
[8]	M. Denker, C. Grillenberger and K. Sigmund, Ergodic Theory on Compact Spaces, Lecture Notes in Mathematics, 527, Springer-Verlag, 1976.
[9]	F. Dyson, Existence of a phase-transition in a one-dimesional Ising ferromagnet, Communications in Mathematical Physics, 12 (1969), 91-107. doi: 10.1007/BF01645907.
[10]	P. Ferrero and B. Schmitt, Théorème de Ruelle-Perron-Frobenius et Métriques Projectives, 1979.
[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), 87-101. doi: 10.1007/BF01208373.
[12]	D. Hilbert, Ueber die Gerade Linie als körzeste Verbindung zweier Punkte, Mathematische Annalen, 46 (1885), 91-96.
[13]	P. Hulse, An example of non-unique $g$-measures, Ergodic Theory and Dynamical Systems, 26 (2006), 439-445. doi: 10.1017/S0143385705000489.
[14]	M. Keane, Strongly Mixing $g$-Measures, Inventiones Mathematicae, 16 (1972), 309-324. doi: 10.1007/BF01425715.
[15]	G. Keller, Equilibrium States in Ergodic Theory, London Mathematical Society, Student Texts, 42, 1998.
[16]	F. Ledrappier, Principe variationnel et systèmes dynamiques symboliques, Zeitschrift für Wahrscheinlichkeitstheorie und verwandte Gebiet, 30 (1974), 185-202. doi: 10.1007/BF00533471.
[17]	C. Liverani, Decay of correlations, Annals of Mathematics, 142 (1995), 239-301. doi: 10.2307/2118636.
[18]	C. Liverani, Decay of correlations for piecewise expanding maps, Journal of Statistical Physics, 78 (1995), 1111-1129. doi: 10.1007/BF02183704.
[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), 1399-1420. doi: 10.1017/S0143385798118023.
[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), P08012.
[21]	K. Marton, Bounding $\bard$-distance by informational divergence: A method to prove measure concentration, Annals of Probability, 24 (1996), 857-866. doi: 10.1214/aop/1039639365.
[22]	K. Marton, Measure concentration for a class of random processes, Probability Theory and Related Fields, 110 (1998), 427-439. doi: 10.1007/s004400050154.
[23]	V. Maume-Deschamps, Correlation decay for Markov maps on a countable state space, Ergodic Theory and Dynamical Systems, 21 (2001), 165-196. doi: 10.1017/S0143385701001110.
[24]	V. Maume-Deschamps, Projective metric and mixing properties on towers, Transactions of the American Mathematical Society, 353 (2001), 3371-3389. doi: 10.1090/S0002-9947-01-02786-6.
[25]	O. Onicescu and G. Mihoc, Sur les Chaînes de variables statistiques, Bulletin de Sciences Mathématiques, 59 (1935), 174-192.
[26]	D. S. Ornstein, An application of ergodic theory to probability theory, The Annals of Probability, 1 (1973), 43-65. doi: 10.1214/aop/1176997024.
[27]	R. Salgado-García and E. Ugalde, Exact scaling in the expansion-modification system, Journal of Statistical Physics, 153 (2013), 842-863. doi: 10.1007/s10955-013-0866-x.
[28]	E. Seneta, Non-negative matrices an Markov Chains, $2^{nd}$ edition, Springer-Verlag, 1973.
[29]	P. Shields, Ergodic Theory of Discrete Sample Paths, Graduate Studies in Mathematics, 13, American Mathematical Society, 1996.
[30]	P. Walters, Ruelle's operator theorem and $g$-measures, Transactions of the American Mathematical Society, 214 (1975), 375-387.

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