Advanced Search
Article Contents
Article Contents

Projective distance and $g$-measures

Abstract Related Papers Cited by
  • 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.
    Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.


    \begin{equation} \\ \end{equation}
  • [1]

    G. Birkhoff, Extensions of Jentzch's theorem, Transactions of the American Mathematical Society, 85 (1957), 219-227.


    M. Bramson and S. Kalikow, Nonuniqueness in $g$-Functions, Israel Journal of Mathematics, 84 (1993), 153-160.doi: 10.1007/BF02761697.


    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.


    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.


    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.


    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.


    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.


    M. Denker, C. Grillenberger and K. Sigmund, Ergodic Theory on Compact Spaces, Lecture Notes in Mathematics, 527, Springer-Verlag, 1976.


    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.


    P. Ferrero and B. SchmittThéorème de Ruelle-Perron-Frobenius et Métriques Projectives, 1979.


    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.


    D. Hilbert, Ueber die Gerade Linie als körzeste Verbindung zweier Punkte, Mathematische Annalen, 46 (1885), 91-96.


    P. Hulse, An example of non-unique $g$-measures, Ergodic Theory and Dynamical Systems, 26 (2006), 439-445.doi: 10.1017/S0143385705000489.


    M. Keane, Strongly Mixing $g$-Measures, Inventiones Mathematicae, 16 (1972), 309-324.doi: 10.1007/BF01425715.


    G. Keller, Equilibrium States in Ergodic Theory, London Mathematical Society, Student Texts, 42, 1998.


    F. Ledrappier, Principe variationnel et systèmes dynamiques symboliques, Zeitschrift für Wahrscheinlichkeitstheorie und verwandte Gebiet, 30 (1974), 185-202.doi: 10.1007/BF00533471.


    C. Liverani, Decay of correlations, Annals of Mathematics, 142 (1995), 239-301.doi: 10.2307/2118636.


    C. Liverani, Decay of correlations for piecewise expanding maps, Journal of Statistical Physics, 78 (1995), 1111-1129.doi: 10.1007/BF02183704.


    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.


    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.


    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.


    K. Marton, Measure concentration for a class of random processes, Probability Theory and Related Fields, 110 (1998), 427-439.doi: 10.1007/s004400050154.


    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.


    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.


    O. Onicescu and G. Mihoc, Sur les Chaînes de variables statistiques, Bulletin de Sciences Mathématiques, 59 (1935), 174-192.


    D. S. Ornstein, An application of ergodic theory to probability theory, The Annals of Probability, 1 (1973), 43-65.doi: 10.1214/aop/1176997024.


    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.


    E. Seneta, Non-negative matrices an Markov Chains, $2^{nd}$ edition, Springer-Verlag, 1973.


    P. Shields, Ergodic Theory of Discrete Sample Paths, Graduate Studies in Mathematics, 13, American Mathematical Society, 1996.


    P. Walters, Ruelle's operator theorem and $g$-measures, Transactions of the American Mathematical Society, 214 (1975), 375-387.

  • 加载中

Article Metrics

HTML views() PDF downloads(125) Cited by(0)

Access History

Other Articles By Authors



    DownLoad:  Full-Size Img  PowerPoint