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Ergodic transport theory and piecewise analytic subactions for analytic dynamics

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Abstract

We consider a piecewise analytic real expanding map f: [0, 1] → [0, 1] of degree d which preserves orientation, and a real analytic positive potential g: [0, 1] → ℝ. We assume the map and the potential have a complex analytic extension to a neighborhood of the interval in the complex plane. We also assume log g is well defined for this extension.

It is known in Complex Dynamics that under the above hypothesis, for the given potential β log g, where β is a real constant, there exists a real analytic eigenfunction ϕ β defined on [0, 1] (with a complex analytic extension) for the Ruelle operator of β log g. Under some assumptions we show that \(\frac{1} {\beta }\log \varphi _\beta\) converges and is a piecewise analytic calibrated subaction.

Our theory can be applied when log g(x) = −log f′(x). In that case we relate the involution kernel to the so called scaling function.

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References

  1. Bamón Rodrigo, Kiwi Jan, Rivera-Letelier Juan and Urzúa Richard. On the topology of solenoidal attractors of the cylinder. Ann. Inst. H. Poincaré Anal. Non Linéaire, 23(2) (2006), 209–236.

    Article  MATH  Google Scholar 

  2. A. Baraviera, A.O. Lopes and P. Thieullen. A large deviation principle for equilibrium states of Hölder potencials: the zero temperature case. Stochastics and Dynamics, 6 (2006), 77–96.

    Article  MathSciNet  MATH  Google Scholar 

  3. A.T. Baraviera, L.M. Cioletti, A.O. Lopes, J. Mohr and R.R. Souza. On the general XY Model: positive and zero temperature, selection and non-selection. Reviews in Math. Physics, 23(10) (2011), 1063–1113.

    Article  MathSciNet  MATH  Google Scholar 

  4. A. Baraviera, R. Leplaideur and A.O. Lopes. Selection of measures for a potential with two maxima at the zero temperature limit. SIAM Journ. on Applied Dynamics, 11(1) (2012), 243–260.

    Article  MathSciNet  MATH  Google Scholar 

  5. P. Bhattacharya and M. Majumdar. Random Dynamical Systems. Cambridge Univ. Press (2007).

  6. T. Bousch. Le poisson n’a pas d’arêtes. Annales de l’Institut Henri Poincaré, Probabilités et Statistiques, 36 (2000), 489–508.

    Article  MathSciNet  MATH  Google Scholar 

  7. T. Bousch. La condition de Walters. Ann. Sci. ENS, 34 (2001), 287–311.

    MathSciNet  MATH  Google Scholar 

  8. T. Bousch and O. Jenkinson. Cohomology classes of dynamically non-negative C k functions. Invent. Math., 148(1) (2002), 207–217.

    Article  MathSciNet  MATH  Google Scholar 

  9. R. Bowen. Equilibrium states and the ergodic theory of Anosov diffeomorphisms. Lecture Notes in Math, 470 (1975).

  10. S. Boyd. Convex Optimization. Cambrige Press (2004).

  11. X. Bressaud and A. Quas. Rate of approximation of minimizing measures. Nonlinearity, 20(4) (2007), 845–853.

    Article  MathSciNet  MATH  Google Scholar 

  12. J.R. Chazottes and M. Hochman. On the zero-temperature limit of Gibbs states. Comm. Math. Phys., 297(1) (2010), 265–281 (2009).

    Article  MathSciNet  MATH  Google Scholar 

  13. W. Chou and R. J. Duffin. An additive eigenvalue problem of physics related to linear programming. Advances in Applied Mathematics, 8 (1987), 486–498.

    Article  MathSciNet  MATH  Google Scholar 

  14. J. Conway. Functions of one complex variable. Springer Verlag (1978).

  15. P. Collet P.J. Lebowitz and A. Porzio. The dimension spectrum of some dynamical systems. J. Stat. Phys., 47 (1984), 609–644.

    Article  MathSciNet  Google Scholar 

  16. G. Contreras and R. Iturriaga. Global Minimizers of Autonomous Lagrangians, 2004, To appear.

  17. G. Contreras, A.O. Lopes and Ph. Thieullen. Lyapunov minimizing measures for expanding maps of the circle. Ergodic Theory and Dynamical Systems, 21 (2001), 1379–1409.

    Article  MathSciNet  MATH  Google Scholar 

  18. G. Contreras, A.O. Lopes and E.R. Oliveira. Ergodic Transport Theory, periodic maximizing probabilities and the twist condition, preprint UFRGS (2011).

  19. J. Delon, J. Salomon and A. Sobolevski. Fast transport optimization for Monge costs on the circle. SIAM J. Appl. Math., 7 (2010), 2239–2258.

    Article  MathSciNet  Google Scholar 

  20. E. de Faria and W. de Melo. Mathematical Tools for one-dimensonal dynamics. Cambridre Press (2008).

  21. W. de Melo and S. Van Strien. One-Dimensional Dynamics. Springer Verlag, (1996).

  22. J.P. Conze and Y. Guivarc’h. Croissance des sommes ergodiques et principe variationnel, manuscript circa (1993).

  23. A. Dembo and O. Zeitouni. Large Deviation Techniques and Applications. Springer Verlag (1998).

  24. R.S. Ellis. Entropy, Large Deviation and Statistical Mechanics. Springer Verlag (1985).

  25. A. Fathi. Weak KAM Theorem and Lagrangian Dynamics, (2004), to appear.

  26. C. Gole. Symplectic Twist Maps. World Scientific (2001).

  27. E. Garibaldi and A.O. Lopes. Functions for relative maximization. Dynamical Systems, 22 (2007), 511–528.

    MathSciNet  MATH  Google Scholar 

  28. E. Garibaldi and A.O. Lopes. On the Aubry-Mather Theory for Symbolic Dynamics. Erg. Theo. and Dyn. Systems, 28(3) (2008), 791–815.

    MathSciNet  MATH  Google Scholar 

  29. E. Garibaldi, A.O. Lopes and Ph. Thieullen. On calibrated and separating subactions. Bull. Braz. Math. Soc., 40(4) (2009), 577–602.

    Article  MathSciNet  MATH  Google Scholar 

  30. D.A. Gomes, A.O. Lopes and J. Mohr. The Mather measure and a Large Deviation Principle for the Entropy Penalized Method. Comm. in Contemp. Math., 13(2) (2011), 235–268.

    Article  MathSciNet  MATH  Google Scholar 

  31. G. Gui, Y. Jiang and A. Quas. Scaling functions, Gibbs measures, and Teichmüller spaces of circle endomorphisms. Discrete Contin. Dynam. Systems, 5(3) (1999), 535–552.

    Article  MathSciNet  Google Scholar 

  32. B.R. Hunt and G.C. Yuan. Optimal orbits of hyperbolic systems. Nonlinearity, 12 (1999), 1207–1224.

    Article  MathSciNet  MATH  Google Scholar 

  33. H.G. Hentschell and I. Proccacia. The infinite number of generalized dimension of fractal and strange attractors. Physica, 8D (1973), 435–44.

    Google Scholar 

  34. O. Jenkinson. Ergodic optimization. Discrete and Continuous Dynamical Systems, Series A, 15 (2006), 197–224.

    Article  MathSciNet  MATH  Google Scholar 

  35. O. Jenkinson and J. Steel. Majorization of invariant measures for orientationreversing maps. Erg. Theo. and Dyn. Syst., (2009).

  36. R. Leplaideur. A dynamical proof for the convergence of Gibbs measures at temperature zero. Nonlinearity, 18(6) (2005), 2847–2880.

    Article  MathSciNet  MATH  Google Scholar 

  37. A.O. Lopes, J. Mohr, R.R. Souza and P. Thieullen. Negative Entropy, Zero temperature and stationary Markov chains on the interval. Bull. Soc. Bras. Math., 40(1) (2009), 1–52.

    Article  MathSciNet  MATH  Google Scholar 

  38. A.O. Lopes. The Dimension Spectrum of the Maximal Measure. SIAM Journal of Mathematical Analysis, 20(5) (1989), 1243–1254.

    Article  MathSciNet  MATH  Google Scholar 

  39. A.O. Lopes. Entropy and Large Deviation. Nonlinearity, 3(2) (1990), 527–546.

    Article  MathSciNet  MATH  Google Scholar 

  40. A.O. Lopes, E.R. Oliveira and P. Thieullen. The dual potential, the involution kernel and transport in ergodic optimization, preprint (2008).

  41. A.O. Lopes and E.R. Oliveira. On the thin boundary of the fat attractor, preprint (2012).

  42. R. Ma.é. Ergodic Theory and Differentiable Dynamics. Springer Verlag (1987).

  43. R. Ma.é. Generic properties and problems of minimizing measures of Lagrangian systems. Nonlinearity, 9 (1996), 273–310.

    Article  MathSciNet  Google Scholar 

  44. M. Martens and W. Melo. The multipliers of periodic points in one-dimensional dynamics. Nonlinearity, 12 (1999), 217–227.

    Article  MathSciNet  MATH  Google Scholar 

  45. J. Mather. Action minimizing invariant measures for positive definite Lagrangian systems. Math. Z., 2 (1991), 169–207.

    Article  MathSciNet  Google Scholar 

  46. J. Milnor. Dynamics in One Complex Variable. Princeton Press (2006).

  47. T. Mitra. Introduction to Dynamic Optimization Theory. Optimization and Chaos, Editors: M. Majumdar, T. Mitra and K. Nishimura, Studies in Economic Theory, Springer Verlag (2000).

  48. I.D. Morris. A sufficient condition for the subordination principle in ergodic optimization. Bull. Lond. Math. Soc., 39(2) (2007), 214–220.

    Article  MathSciNet  MATH  Google Scholar 

  49. L. Olsen. A multifractal formalism. Adv. Math., 116(1) (1995), 82–196.

    Article  MathSciNet  MATH  Google Scholar 

  50. W. Parry and M. Pollicott. Zeta functions and the periodic orbit structure of hyperbolic dynamics. Astérisque, 187–188 (1990).

  51. A.A. Pinto and D. Rand. Existence, uniqueness and ratio decomposition for Gibbs states via duality. Ergodic Theory Dynam. Systems, 21(2) (2001), 533–543.

    Article  MathSciNet  MATH  Google Scholar 

  52. M. Pollicott. Some applications of thermodynamic formalism to manifolds with constant negative curvature. Adv. Math., 85(2) (1991), 161–192.

    Article  MathSciNet  MATH  Google Scholar 

  53. M. Pollicott. Symbolic dynamics and geodesic flows. Séminaire de Théorie Spectrale et Géométrie, No. 10, Année 1991–1992, 109–129, Univ. Grenoble I, Saint-Martin-d’Hères (1992).

  54. D. Ruelle. Repellers for real analytic maps. Ergodic Theory Dynamical Systems, 2(1) (1982), 99–107.

    Article  MathSciNet  MATH  Google Scholar 

  55. F. Przytycki and M. Urbanski. Conformal Fractals: Ergodic Theory Methods. Cambridge Press (2010).

  56. M. Shub and D. Sullivan. Expanding endomorphisms of the circle revisited. Ergodic Theory Dynam. Systems, 5(2) (1985), 285–289.

    Article  MathSciNet  MATH  Google Scholar 

  57. F.A. Tal and S.A. Zanata. Maximizing measures for endomorphisms of the circle. Nonlinearity, 21 (2008).

  58. M. Tsujii. Fat solenoidal attractors. Nonlinearity, 14 (2001), 1011–1027.

    Article  MathSciNet  MATH  Google Scholar 

  59. C. Villani. Topics in optimal transportation. AMS, Providence (2003).

    MATH  Google Scholar 

  60. C. Villani. Optimal transport: old and new. Springer-Verlag, Berlin (2009).

    Book  MATH  Google Scholar 

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Author information

Authors and Affiliations

  1. Instituto de Matemática, UFRGS, 91509-900, Porto Alegre, RS, Brazil

    A. O. Lopes & E. R. Oliveira

  2. Departamento de Matemática, ICMC-USP, 13560-970, São Carlos, SP, Brazil

    D. Smania

Authors
  1. A. O. Lopes
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  2. E. R. Oliveira
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  3. D. Smania
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Correspondence to A. O. Lopes.

Additional information

Partially supported by CNPq, PRONEX — Sistemas Dinamicos, Instituto do Milênio, and beneficiary of CAPES financial support.

Partially supported by CNPq 310964/2006-7, and FAPESP 2008/02841-4.

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Lopes, A.O., Oliveira, E.R. & Smania, D. Ergodic transport theory and piecewise analytic subactions for analytic dynamics. Bull Braz Math Soc, New Series 43, 467–512 (2012). https://doi.org/10.1007/s00574-012-0023-1

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  • DOI: https://doi.org/10.1007/s00574-012-0023-1

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