Advertisement

Basic Notions and Examples

  • Luis Barreira
Chapter
Part of the Universitext book series (UTX)

Abstract

We introduce in this chapter the basic notions of hyperbolic dynamics, starting with the concept of hyperbolicity. We also establish several basic properties of hyperbolic sets, including the continuous dependence of the stable and unstable subspaces on the base point. In addition, we discuss several examples of hyperbolic sets. These include hyperbolic fixed points, the Smale horseshoe, and hyperbolic automorphisms of the 2-torus. We also construct coding maps via symbolic dynamics. Finally, we consider noninvertible transformations and their repellers, and we construct corresponding Markov partitions. We refer to the following chapter for the more elaborate construction of Markov partitions for hyperbolic sets.

Keywords

Periodic Point Negative Curvature Markov Property Symbolic Dynamic Geodesic Flow 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. 1.
    R. Adler, A. Konheim, M. McAndrew, Topological entropy. Trans. Am. Math. Soc. 114, 309–319 (1965)Google Scholar
  2. 2.
    R. Adler, B. Weiss, Similarity of automorphisms of the torus. Mem. Am. Math. Soc. 98 (1970)Google Scholar
  3. 3.
    D. Anosov, Roughness of geodesic flows on compact Riemannian manifolds of negative curvature. Dokl. Akad. Nauk SSSR 145, 707–709 (1962)Google Scholar
  4. 4.
    D. Anosov, Geodesic flows on closed Riemann manifolds with negative curvature. Proc. Steklov Inst. Math. 90, 1–235 (1967)Google Scholar
  5. 5.
    D. Anosov, Ya. Sinai, Certain smooth ergodic systems. Russ. Math. Surv. 22, 103–167 (1967)Google Scholar
  6. 6.
    M. Artin, B. Mazur, On periodic points. Ann. Math. (2) 81, 82–99 (1965)Google Scholar
  7. 7.
    L. Barreira, A non-additive thermodynamic formalism and applications to dimension theory of hyperbolic dynamical systems. Ergod. Theor. Dyn. Syst. 16, 871–927 (1996)Google Scholar
  8. 8.
    L. Barreira, Dimension and Recurrence in Hyperbolic Dynamics. Progress in Mathematics, vol. 272 (Birkhäuser, Basel, 2008)Google Scholar
  9. 9.
    L. Barreira, Thermodynamic Formalism and Applications to Dimension Theory. Progress in Mathematics, vol. 294 (Birkhäuser, Basel, 2011)Google Scholar
  10. 10.
    L. Barreira, Ya. Pesin, Lyapunov Exponents and Smooth Ergodic Theory. University Lecture Series, vol. 23 (American Mathematical Society, RI, 2002)Google Scholar
  11. 11.
    L. Barreira, Ya. Pesin, Nonuniform Hyperbolicity. Encyclopedia of Mathematics and Its Applications, vol. 115 (Cambridge University Press, London, 2007)Google Scholar
  12. 12.
    L. Barreira, Ya. Pesin, J. Schmeling, Dimension and product structure of hyperbolic measures. Ann. Math. (2) 149, 755–783 (1999)Google Scholar
  13. 13.
    L. Barreira, B. Saussol, Hausdorff dimension of measures via Poincaré recurrence. Comm. Math. Phys. 219, 443–463 (2001)Google Scholar
  14. 14.
    L. Barreira, J. Schmeling, Sets of “non-typical” points have full topological entropy and full Hausdorff dimension. Isr. J. Math. 116, 29–70 (2000)Google Scholar
  15. 15.
    G. Birkhoff, Proof of the ergodic theorem. Proc. Acad. Sci. USA 17, 656–660 (1931)Google Scholar
  16. 16.
    M. Boshernitzan, Quantitative recurrence results. Invent. Math. 113, 617–631 (1993)Google Scholar
  17. 17.
    H. Bothe, The Hausdorff dimension of certain solenoids. Ergod. Theor. Dyn. Syst. 15, 449–474 (1995)Google Scholar
  18. 18.
    R. Bowen, Markov partitions for Axiom A diffeomorphisms. Am. J. Math. 92, 725–747 (1970)Google Scholar
  19. 19.
    R. Bowen, Topological entropy and axiom A. In Global Analysis (Proc. Sympos. Pure Math. XIV, Berkeley, 1968) (American Mathematical Society, RI, 1970), pp. 23–41Google Scholar
  20. 20.
    R. Bowen, Equilibrium States and Ergodic Theory of Anosov Diffeomorphisms. Lecture Notes in Mathematics, vol. 470 (Springer, Berlin, 1975)Google Scholar
  21. 21.
    R. Bowen, Hausdorff dimension of quasi-circles. Inst. Hautes Études Sci. Publ. Math. 50, 259–273 (1979)Google Scholar
  22. 22.
    L. Breiman, The individual ergodic theorem of information theory. Ann. Math. Stat. 28, 809–811 (1957)Google Scholar
  23. 23.
    M. Brin, G. Stuck, Introduction to Dynamical Systems (Cambridge University Press, London, 2002)Google Scholar
  24. 24.
    G. Choe, Computational Ergodic Theory. Algorithms and Computation in Mathematics, vol. 13 (Springer, Berlin, 2005)Google Scholar
  25. 25.
    P. Collet, J. Lebowitz, A. Porzio, The dimension spectrum of some dynamical systems. J. Stat. Phys. 47, 609–644 (1987)Google Scholar
  26. 26.
    I. Cornfeld, S. Fomin, Ya. Sinai, Ergodic Theory. Grundlehren der mathematischen Wissenchaften, vol. 245 (Springer, Berlin, 1982)Google Scholar
  27. 27.
    E. Dinaburg, On the relations among various entropy characteristics of dynamical systems. Math. USSR-Izv. 5, 337–378 (1971)Google Scholar
  28. 28.
    A. Douady, J. Oesterlé, Dimension de Hausdorff des attracteurs. C. R. Acad. Sc. Paris 290, 1135–1138 (1980)Google Scholar
  29. 29.
    K. Falconer, The Geometry of Fractal Sets. Cambridge Tracts in Mathematics, vol. 85 (Cambridge University Press, London, 1986)Google Scholar
  30. 30.
    K. Falconer, The Hausdorff dimension of self-affine fractals. Math. Proc. Camb. Phil. Soc. 103, 339–350 (1988)Google Scholar
  31. 31.
    K. Falconer, Dimensions and measures of quasi self-similar sets. Proc. Am. Math. Soc. 106, 543–554 (1989)Google Scholar
  32. 32.
    K. Falconer, Bounded distortion and dimension for non-conformal repellers. Math. Proc. Camb. Phil. Soc. 115, 315–334 (1994)Google Scholar
  33. 33.
    K. Falconer, Fractal Geometry. Mathematical Foundations and Applications (Wiley, NY, 2003)Google Scholar
  34. 34.
    T. Goodman, Relating topological entropy and measure entropy. Bull. Lond. Math. Soc. 3, 176–180 (1971)Google Scholar
  35. 35.
    T. Goodman, Maximal measures for expansive homeomorphisms. J. Lond. Math. Soc. (2) 5, 439–444 (1972)Google Scholar
  36. 36.
    L. Goodwyn, Topological entropy bounds measure-theoretic entropy. Proc. Am. Math. Soc. 23, 679–688 (1969)Google Scholar
  37. 37.
    J. Hadamard, Les surfaces à courbures opposées et leur lignes géodesiques. J. Math. Pure. Appl. 4, 27–73 (1898)Google Scholar
  38. 38.
    T. Halsey, M. Jensen, L. Kadanoff, I. Procaccia, B. Shraiman, Fractal measures and their singularities: The characterization of strange sets. Phys. Rev. A (3) 34, 1141–1151 (1986); errata in 34, 1601 (1986)Google Scholar
  39. 39.
    B. Hasselblatt, Regularity of the Anosov splitting and of horospheric foliations. Ergod. Theor. Dyn. Syst. 14, 645–666 (1994)Google Scholar
  40. 40.
    M. Hirsch, C. Pugh, Stable manifolds and hyperbolic sets. In Global Analysis (Proc. Sympos. Pure Math. XIV, Berkeley, 1968) (American Mathematical Society, RI, 1970), pp. 133–163Google Scholar
  41. 41.
    H. Hu, Box dimensions and topological pressure for some expanding maps. Comm. Math. Phys. 191, 397–407 (1998)Google Scholar
  42. 42.
    M. Irwin, Smooth Dynamical Systems (Academic Press, NY, 1980)Google Scholar
  43. 43.
    A. Katok, Lyapunov exponents, entropy and periodic orbits for diffeomorphisms. Inst. Hautes Études Sci. Publ. Math. 51, 137–173 (1980)Google Scholar
  44. 44.
    A. Katok, B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems. Encyclopedia of Mathematics and Its Applications, vol. 54 (Cambridge University Press, London, 1995)Google Scholar
  45. 45.
    G. Keller, Equilibrium States in Ergodic Theory. London Mathematical Society Student Texts, vol. 42 (Cambridge University Press, London, 1998)Google Scholar
  46. 46.
    A. Khinchin, On the basic theorems of information theory. Uspehi Mat. Nauk (N.S.) 11, 17–75 (1956)Google Scholar
  47. 47.
    A. Khinchin, Mathematical Foundations of Information Theory (Dover, NY, 1957)Google Scholar
  48. 48.
    B. Kitchens, Symbolic Dynamics, One-Sided, Two-Sided and Countable State Markov Shifts. Universitext (Springer, Berlin, 1998)Google Scholar
  49. 49.
    A. Kolmogorov, A new metric invariant of transient dynamical systems and automorphisms in Lebesgue spaces. Dokl. Akad. Nauk SSSR (N.S.) 119, 861–864 (1958)Google Scholar
  50. 50.
    A. Kolmogorov, Entropy per unit time as a metric invariant of automorphisms. Dokl. Akad. Nauk SSSR 124, 754–755 (1959)Google Scholar
  51. 51.
    U. Krengel, Ergodic Theorems (de Gruyter, Berlin, 1985)Google Scholar
  52. 52.
    N. Kryloff, N. Bogoliouboff, La théorie générale de la mesure dans son application à l’étude des systèmes dynamiques de la mécanique non linéaire. Ann. Math. (2) 38, 65–113 (1937)Google Scholar
  53. 53.
    H. Lebesgue, Intégrale, longueur, aire. Ann. Mat. Pura Appl. (3) 7, 231–359 (1902)Google Scholar
  54. 54.
    F. Ledrappier, L.-S. Young, The metric entropy of diffeomorphisms II. Relations between entropy, exponents and dimension. Ann. Math. (2) 122, 540–574 (1985)Google Scholar
  55. 55.
    D. Lind, B. Marcus, An Introduction to Symbolic Dynamics and Coding (Cambridge University Press, London, 1995)Google Scholar
  56. 56.
    A. Lopes, The dimension spectrum of the maximal measure. SIAM J. Math. Anal. 20, 1243–1254 (1989)Google Scholar
  57. 57.
    R. Mañé, Ergodic Theory and Differentiable Dynamics. Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 8 (Springer, Berlin, 1987)Google Scholar
  58. 58.
    P. Mattila, Geometry of Sets and Measures in Euclidean Spaces. Fractals and Rectifiability. Cambridge Studies in Advanced Mathematics, vol. 44 (Cambridge University Press, London, 1995)Google Scholar
  59. 59.
    H. McCluskey, A. Manning, Hausdorff dimension for horseshoes. Ergod. Theor. Dyn. Syst. 3, 251–260 (1983)Google Scholar
  60. 60.
    B. McMillan, The basic theorems of information theory. Ann. Math. Stat. 24, 196–219 (1953)Google Scholar
  61. 61.
    M. Misiurewicz, A short proof of the variational principle for a Z  +  N action on a compact space. In International Conference on Dynamical Systems in Mathematical Physics, Rennes, 1975. Astérisque, vol. 40 (Soc. Math. France, Montrouge, 1976), pp. 147–157Google Scholar
  62. 62.
    P. Moran, Additive functions of intervals and Hausdorff measure. Proc. Camb. Phil. Soc. 42, 15–23 (1946)Google Scholar
  63. 63.
    M. Morse, A one-to-one representation of geodesics on a surface of negative curvature. Am. J. Math. 43, 33–51 (1921)Google Scholar
  64. 64.
    M. Morse, G. Hedlund, Symbolic dynamics. Am. J. Math. 60, 815–866 (1938)Google Scholar
  65. 65.
    Z. Nitecki, Differentiable Dynamics (MIT Press, MA, 1971)Google Scholar
  66. 66.
    D. Ornstein, Ergodic Theory, Randomness, and Dynamical Systems. Yale Mathematical Monographs, vol. 5 (Yale University Press, CT, 1974)Google Scholar
  67. 67.
    D. Ornstein, B. Weiss, Entropy and data compression schemes. IEEE Trans. Inform. Theor. 39, 78–83 (1993)Google Scholar
  68. 68.
    J. Palis, M. Viana, On the continuity of Hausdorff dimension and limit capacity for horseshoes. In Dynamical Systems (Valparaiso, 1986), ed. by R. Bamón, R. Labarca, J. Palis. Lecture Notes in Mathematics, vol. 1331 (Springer, Berlin, 1988), pp. 150–160Google Scholar
  69. 69.
    W. Parry, Topics in Ergodic Theory (Cambridge University Press, London, 1981)Google Scholar
  70. 70.
    W. Parry, M. Pollicott, Zeta Functions and the Periodic Orbit Structure of Hyperbolic Dynamics. Astérisque, vol. 187–188 (Soc. Math. France, Montrouge, 1990)Google Scholar
  71. 71.
    O. Perron, Die Stabilitätsfrage bei Differentialgleichungen. Math. Z. 32, 703–728 (1930)Google Scholar
  72. 72.
    Ya. Pesin, Characteristic exponents and smooth ergodic theory. Russ. Math. Surv. 32, 55–114 (1977)Google Scholar
  73. 73.
    Ya. Pesin, Dimension Theory in Dynamical Systems. Contemporary Views and Applications. Chicago Lectures in Mathematics (Chicago University Press, IL, 1997)Google Scholar
  74. 74.
    Ya. Pesin, B. Pitskel’, Topological pressure and the variational principle for noncompact sets. Funct. Anal. Appl. 18, 307–318 (1984)Google Scholar
  75. 75.
    Ya. Pesin, H. Weiss, On the dimension of deterministic and random Cantor-like sets, symbolic dynamics, and the Eckmann–Ruelle conjecture. Comm. Math. Phys. 182, 105–153 (1996)Google Scholar
  76. 76.
    Ya. Pesin, H. Weiss, A multifractal analysis of Gibbs measures for conformal expanding maps and Markov Moran geometric constructions. J. Stat. Phys. 86, 233–275 (1997)Google Scholar
  77. 77.
    K. Petersen, Ergodic Theory (Cambridge University Press, London, 1983)Google Scholar
  78. 78.
    H. Poincaré, Sur le problème des trois corps et les équations de la dynamique. Acta Math. 13, 1–270 (1890)Google Scholar
  79. 79.
    M. Pollicott, M. Yuri, Dynamical Systems and Ergodic Theory. London Mathematical Society Student Texts, vol. 40 (Cambridge University Press, London, 1998)Google Scholar
  80. 80.
    D. Rand, The singularity spectrum f(α) for cookie-cutters. Ergod. Theor. Dyn. Syst. 9, 527–541 (1989)Google Scholar
  81. 81.
    C. Robinson, Dynamical Systems. Stability, Symbolic Dynamics, and Chaos. Studies in Advanced Mathematics (CRC Press, FL, 1995)Google Scholar
  82. 82.
    D. Rudolph, Fundamentals of Measurable Dynamics: Ergodic Theory on Lebesgue Spaces (Oxford University Press, London, 1990)Google Scholar
  83. 83.
    D. Ruelle, Statistical mechanics on a compact set with ν action satisfying expansiveness and specification. Trans. Am. Math. Soc. 185, 237–251 (1973)Google Scholar
  84. 84.
    D. Ruelle, Thermodynamic Formalism. Encyclopedia of Mathematics and Its Applications, vol. 5 (Addison-Wesley, MA, 1978)Google Scholar
  85. 85.
    D. Ruelle, Repellers for real analytic maps. Ergod. Theor. Dyn. Syst. 2, 99–107 (1982)Google Scholar
  86. 86.
    J. Schmeling, On the completeness of multifractal spectra. Ergod. Theor. Dyn. Syst. 19, 1595–1616 (1999)Google Scholar
  87. 87.
    C. Shannon, A mathematical theory of communication. Bell Syst. Tech. J. 27, 379–423, 623–656 (1948)Google Scholar
  88. 88.
    M. Shereshevsky, A complement to Young’s theorem on measure dimension: The difference between lower and upper pointwise dimension. Nonlinearity 4, 15–25 (1991)Google Scholar
  89. 89.
    M. Shub, Global Stability of Dynamical Systems (Springer, Berlin, 1986)Google Scholar
  90. 90.
    K. Simon, The Hausdorff dimension of the Smale–Williams solenoid with different contraction coefficients. Proc. Am. Math. Soc. 125, 1221–1228 (1997)Google Scholar
  91. 91.
    K. Simon, B. Solomyak, Hausdorff dimension for horseshoes in 3. Ergod. Theor. Dyn. Syst. 19, 1343–1363 (1999)Google Scholar
  92. 92.
    D. Simpelaere, Dimension spectrum of axiom A diffeomorphisms. II. Gibbs measures. J. Stat. Phys. 76, 1359–1375 (1994)Google Scholar
  93. 93.
    Ya. Sinai, On the concept of entropy for a dynamic system. Dokl. Akad. Nauk SSSR 124, 768–771 (1959)Google Scholar
  94. 94.
    Ya. Sinai, Construction of Markov partitions. Funct. Anal. Appl. 2, 245–253 (1968a)Google Scholar
  95. 95.
    Ya. Sinai, Markov partitions and C-diffeomorphisms. Funct. Anal. Appl. 2, 61–82 (1968b)Google Scholar
  96. 96.
    Ya. Sinai, Introduction to Ergodic Theory. Mathematical Notes, vol. 18 (Princeton University Press, NJ, 1976)Google Scholar
  97. 97.
    Ya. Sinai, Topics in Ergodic Theory. Princeton Mathematical Series, vol. 44 (Princeton University Press, NJ, 1994)Google Scholar
  98. 98.
    S. Smale, Diffeomorphisms with many periodic points. In Differential and Combinatorial Topology (A Symposium in Honor of Marston Morse) (Princeton University Press, NJ, 1965), pp. 63–80Google Scholar
  99. 99.
    S. Smale, Differentiable dynamical systems. Bull. Am. Math. Soc. 73, 747–817 (1967)Google Scholar
  100. 100.
    W. Szlenk, An Introduction to the Theory of Smooth Dynamical Systems (Wiley, NY, 1984)Google Scholar
  101. 101.
    F. Takens, Limit capacity and Hausdorff dimension of dynamically defined Cantor sets. In Dynamical Systems (Valparaiso, 1986), ed. by R. Bamón, R. Labarca, J. Palis. Lecture Notes in Mathematics, vol. 1331 (Springer, Berlin, 1988), pp. 196–212Google Scholar
  102. 102.
    P. Walters, A variational principle for the pressure of continuous transformations. Am. J. Math. 97, 937–971 (1976)Google Scholar
  103. 103.
    P. Walters, An Introduction to Ergodic Theory. Graduate Texts in Mathematics, vol. 79 (Springer, Berlin, 1982)Google Scholar
  104. 104.
    H. Weyl, Über der Gleichverteilung von Zahlen mod Eins. Math. Ann. 77, 313–352 (1916)Google Scholar
  105. 105.
    L.-S. Young, Dimension, entropy and Lyapunov exponents. Ergod. Theor. Dyn. Syst. 2, 109–124 (1982)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Luis Barreira
    • 1
  1. 1.Departamento de MatemáticaInstituto Superior TécnicoLisboaPortugal

Personalised recommendations