Mathematics of Complexity and Dynamical Systems

2011 Edition
| Editors: Robert A. Meyers (Editor-in-Chief)

Ergodic Theory: Fractal Geometry

  • Jörg Schmeling
Reference work entry
DOI: https://doi.org/10.1007/978-1-4614-1806-1_18

Article Outline

Glossary

Definition of the Subject

Introduction

Preliminaries

Brief Tour Through Some Examples

Dimension Theory of Low-Dimensional Dynamical Systems – Young's Dimension Formula

Dimension Theory of Higher-Dimensional Dynamical Systems

Hyperbolic Measures

General Theory

Endomorphisms

Multifractal Analysis

Future Directions

Bibliography

Keywords

Lyapunov Exponent Invariant Measure Hausdorff Dimension Rotation Number Topological Entropy 
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.
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Bibliography

Primary Literature

  1. 1.
    Afraimovich VS, Schmeling J, Ugalde J, Urias J (2000) Spectra of dimension for Poincaré recurrences. Discret Cont Dyn Syst 6(4):901–914MathSciNetMATHGoogle Scholar
  2. 2.
    Afraimovich VS, Chernov NI, Sataev EA (1995) Statistical Properties of 2D Generalized Hyperbolic Attractors. Chaos 5:238–252MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Aihua F, Yunping J, Jun W (2005) Asymptotic Hausdorff dimensions of Cantor sets associated with an asymptotically non‐hyperbolic family. Ergod Theory Dynam Syst 25(6):1799–1808MATHCrossRefGoogle Scholar
  4. 4.
    Alexander J, Yorke J (1984) Fat Baker's transformations. Erg Th Dyn Syst 4:1–23MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Ambroladze A, Schmeling J (2004) Lyapunov exponents are not stable with respect to arithmetic subsequences. In: Fractal geometry and stochastics III. Progr Probab 57. Birkhäuser, Basel, pp 109–116CrossRefGoogle Scholar
  6. 6.
    Artin E (1965) Ein mechanisches System mit quasi‐ergodischen Bahnen, Collected papers. Addison Wesley, pp 499–501Google Scholar
  7. 7.
    Barreira L () Variational properties of multifractal spectra. IST preprintGoogle Scholar
  8. 8.
    Barreira L (1995) Cantor sets with complicated geometry and modeled by general symbolic dynamics. Random Comp Dyn 3:213–239MathSciNetMATHGoogle Scholar
  9. 9.
    Barreira L (1996) A non-additive thermodynamic formalism and applications to dimension theory of hyperbolic dynamical systems. Erg Th Dyn Syst 16:871–927MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Barreira L (1996) A non-additive thermodynamic formalism and dimension theory of hyperbolic dynamical systems. Math Res Lett 3:499–509MathSciNetMATHGoogle Scholar
  11. 11.
    Barreira L, Saussol B (2001) Hausdorff dimension of measure via Poincaré recurrence. Comm Math Phys 219(2):443–463MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    Barreira L, Saussol B (2001) Multifractal analysis of hyperbolic flows. Comm Math Phys 219(2):443–463MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    Barreira L, Saussol B (2001) Variational principles and mixed multifractal spectra. Trans Amer Math Soc 353(10):3919–3944 (electronic)MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    Barreira L, Schmeling J (2000) Sets of “Non‐typical” Points Have Full Topological Entropy and Full Hausdorff Dimension. Isr J Math 116:29–70MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    Barreira L, Pesin Y, Schmeling J (1997) Multifractal spectra and multifractal rigidity for horseshoes. J Dyn Contr Syst 3:33–49MathSciNetMATHCrossRefGoogle Scholar
  16. 16.
    Barreira L, Pesin Y, Schmeling J (1997) On a General Concept of Multifractal Rigidity: Multifractal Spectra For Dimensions, Entropies, and Lyapunov Exponents. Multifractal Rigidity. Chaos 7:27–38MathSciNetMATHCrossRefGoogle Scholar
  17. 17.
    Barreira L, Pesin Y, Schmeling J (1999) Dimension and Product Structure of Hyperbolic Measures. Annals Math 149:755–783MathSciNetMATHCrossRefGoogle Scholar
  18. 18.
    Barreira L, Saussol B, Schmeling J (2002) Distribution of frequencies of digits via multifractal analysis. J Number Theory 97(2):410–438MathSciNetMATHCrossRefGoogle Scholar
  19. 19.
    Barreira L, Saussol B, Schmeling J (2002) Higher–dimensional multifractal analysis. J Math Pures Appl (9) 81(1):67–91MathSciNetGoogle Scholar
  20. 20.
    Belykh VP (1982) Models of discrete systems of phase synchronization. In: Shakhildyan VV, Belynshina LN (eds) Systems of Phase Synchronization. Radio i Svyaz, Moscow, pp 61–176Google Scholar
  21. 21.
    Billingsley P (1978) Ergodic Theory and Information. KriegerGoogle Scholar
  22. 22.
    Blinchevskaya M, Ilyashenko Y (1999) Estimate for the Entropy Dimension Of The Maximal Attractor For \({k-}\)Constracting Systems In An Infinite‐Dimensional Space. Russ J Math Phys 6(1):20–26MathSciNetMATHGoogle Scholar
  23. 23.
    Boshernitzan M (1993) Quantitative recurrence results. Invent Math 113:617–631MathSciNetMATHCrossRefGoogle Scholar
  24. 24.
    Bothe H-G (1995) The Hausdorff dimension of certain solenoids. Erg Th Dyn Syst 15:449–474MathSciNetMATHCrossRefGoogle Scholar
  25. 25.
    Bousch T (2000) Le poisson n'a pas d'arêtes. Ann IH Poincaré (Prob-Stat) 36(4):489–508MathSciNetMATHCrossRefGoogle Scholar
  26. 26.
    Bowen R (1973) Topological entropy for noncompact sets. Trans Amer Math Soc 184:125–136MathSciNetCrossRefGoogle Scholar
  27. 27.
    Bowen R (1979) Hausdorff Dimension Of Quasi‐circles. Publ Math IHES 50:11–25MathSciNetMATHGoogle Scholar
  28. 28.
    Bylov D, Vinograd R, Grobman D, Nemyckii V (1966) Theory of Lyapunov exponents and its application to problems of stability. Izdat “Nauka”, Moscow (in Russian)Google Scholar
  29. 29.
    Casdagli M, Sauer T, Yorke J (1991) Embedology. J Stat Phys 65:589–616MathSciNetGoogle Scholar
  30. 30.
    Ciliberto S, Eckmann JP, Kamphorst S, Ruelle D (1971) Liapunov Exponents from Times. Phys Rev A 34Google Scholar
  31. 31.
    Collet P, Lebowitz JL, Porzio A (1987) The Dimension Spectrum of Some Dynamical Systems. J Stat Phys 47:609–644MathSciNetMATHCrossRefGoogle Scholar
  32. 32.
    Constantin P, Foias C (1988) Navier‐Stokes Equations. Chicago U PressGoogle Scholar
  33. 33.
    Cruchfield J, Farmer D, Packard N, Shaw R (1980) Geometry from a Time Series. Phys Rev Lett 45:712–724CrossRefGoogle Scholar
  34. 34.
    Cutler C (1990) Connecting Ergodicity and Dimension in Dynamical Systems. Ergod Th Dynam Syst 10:451–462MathSciNetMATHCrossRefGoogle Scholar
  35. 35.
    Denjoy A (1932) Sur les courbes défines par les équations différentielles á la surface du tore. J Math Pures Appl 2:333–375Google Scholar
  36. 36.
    Ding M, Grebogi C, Ott E, Yorke J (1993) Estimating correlation dimension from a chaotic times series: when does the plateau onset occur? Phys D 69:404–424MathSciNetMATHCrossRefGoogle Scholar
  37. 37.
    Dodson M, Rynne B, Vickers J (1990) Diophantine approximation and a lower bound for Hausdorff dimension. Mathematika 37:59–73MathSciNetMATHCrossRefGoogle Scholar
  38. 38.
    Douady A, Oesterle J (1980) Dimension de Hausdorff Des Attracteurs. CRAS 290:1135–1138MathSciNetMATHGoogle Scholar
  39. 39.
    Eggleston HG (1952) Sets of Fractional Dimension Which Occur in Some Problems of Number Theory. Proc Lond Math Soc 54:42–93MathSciNetCrossRefGoogle Scholar
  40. 40.
    Ellis R (1984) Large Deviations for a General Class of Random Vectors. Ann Prob 12:1–12MATHCrossRefGoogle Scholar
  41. 41.
    Ellis R (1985) Entropy, Large Deviations, and Statistical Mechanics. SpringerGoogle Scholar
  42. 42.
    Falconer K (1990) Fractal Geometry, Mathematical Foundations and Applications. Cambridge U Press, CambridgeMATHGoogle Scholar
  43. 43.
    Fan AH, Feng DJ, Wu J (2001) Recurrence, dimension and entropy. J Lond Math Soc 64(2):229–244MathSciNetMATHCrossRefGoogle Scholar
  44. 44.
    Frederickson P, Kaplan J, Yorke E, Yorke J (1983) The Liapunov Dimension Of Strange Attractors. J Differ Equ 49:185–207MathSciNetMATHCrossRefGoogle Scholar
  45. 45.
    Frostman O (1935) Potential d'équilibre Et Capacité des Ensembles Avec Quelques Applications à la Théorie des Fonctions. Meddel Lunds Univ Math Sem 3:1–118Google Scholar
  46. 46.
    Furstenberg H (1967) Disjointness in Ergodic Theory, Minimal Sets, and a Problem in Diophantine Approximation. Math Syst Theory 1:1–49MathSciNetMATHCrossRefGoogle Scholar
  47. 47.
    Furstenberg H (1970) Intersections of Cantor Sets and Transversality of Semigroups I. In: Problems in Analysis. Sympos Salomon Bochner, Princeton Univ. Princeton Univ Press, pp 41–59Google Scholar
  48. 48.
    Gatzouras D, Peres Y (1996) The variational principle for Hausdorff dimension: A survey, in Ergodic theory of \({\mathbb{Z}^d}\) actions. In: Pollicott M et al (ed) Proc of the Warwick symposium. Cambridge University Press. Lond Math Soc Lect Note Ser 228:113–125MathSciNetGoogle Scholar
  49. 49.
    Grassberger P, Procaccia I, Hentschel H (1983) On the Characterization of Chaotic Motions, Lect Notes. Physics 179:212–221Google Scholar
  50. 50.
    Halsey T, Jensen M, Kadanoff L, Procaccia I, Shraiman B (1986) Fractal Measures and Their Singularities: The Characterization of Strange Sets. Phys Rev A 33(N2):1141–1151MathSciNetMATHCrossRefGoogle Scholar
  51. 51.
    Hasselblatt B (1994) Regularity of the Anosov splitting and of horospheric foliations. Ergod Theory Dynam Syst 14(4):645–666MathSciNetMATHCrossRefGoogle Scholar
  52. 52.
    Hasselblatt B, Schmeling J (2004) Dimension product structure of hyperbolic sets. In: Modern dynamical systems and applications. Cambridge Univ Press, Cambridge, pp 331–345Google Scholar
  53. 53.
    Henley D (1992) Continued Fraction Cantor Sets, Hausdorff Dimension, and Functional Analysis. J Number Theory 40:336–358MathSciNetCrossRefGoogle Scholar
  54. 54.
    Hentschel HGE, Procaccia I (1983) The Infinite Number of Generalized Dimensions of Fractals and Strange Attractors. Physica 8D:435–444MathSciNetGoogle Scholar
  55. 55.
    Herman MR (1979) Sur la conjugaison différentiable des difféomorphismes du cercle á des rotations. Publications de l'Institute de Mathématiques des Hautes Études Scientifiques 49:5–234Google Scholar
  56. 56.
    Hunt B (1996) Maximal Local Lyapunov Dimension Bounds The Box Dimension Of Chaotic Attractors. Nonlinearity 9:845–852MathSciNetMATHCrossRefGoogle Scholar
  57. 57.
    Iommi G (2005) Multifractal analysis for countable Markov shifts. Ergod Theory Dynam Syst 25(6):1881–1907MathSciNetMATHCrossRefGoogle Scholar
  58. 58.
    Jarnik V (1931) Über die simultanen diophantischen Approximationen. Math Zeitschr 33:505–543MathSciNetCrossRefGoogle Scholar
  59. 59.
    Jenkinson O (2001) Rotation, entropy, and equilibrium states. Trans Amer Math Soc 353:3713–3739MathSciNetMATHCrossRefGoogle Scholar
  60. 60.
    Kalinin B, Sadovskaya V (2002) On pointwise dimension of non‐hyperbolic measures. Ergod Theory Dynam Syst 22(6):1783–1801MathSciNetMATHCrossRefGoogle Scholar
  61. 61.
    Kaplan JL, Yorke JA (1979) Functional differential equations and approximation of fixed points. Lecture Notes. In: Mathematics vol 730. Springer, Berlin, pp 204–227Google Scholar
  62. 62.
    Katznelson Y, Weiss B (1982) A simple proof of some ergodic theorems. Isr J of Math 42:291–296MathSciNetMATHCrossRefGoogle Scholar
  63. 63.
    Kenyon R, Peres Y (1996) Measure of full dimension on affine invariant sets. Erg Th Dyn Syst 16:307–323MathSciNetMATHCrossRefGoogle Scholar
  64. 64.
    Kesseböhmer M (1999) Multifrakale und Asymptotiken grosser Deviationen. Thesis U Göttingen, GöttingenGoogle Scholar
  65. 65.
    Kingman JFC (1968) The ergodic theory of subadditive stochastic processes. J Royal Stat Soc B30:499–510MathSciNetMATHGoogle Scholar
  66. 66.
    Kleinbock D, Margulis G (1998) Flows on Homogeneous Spaces and Diophantine Approximation on Manifold. Ann Math 148:339–360MathSciNetMATHCrossRefGoogle Scholar
  67. 67.
    Kra B, Schmeling J (2002) Diophantine classes, dimension and Denjoy maps. Acta Arith 105(4):323–340MathSciNetMATHCrossRefGoogle Scholar
  68. 68.
    Ledrappier F (1981) Some Relations Between Dimension And Lyapounov Exponents. Comm Math Phys 81:229–238MathSciNetMATHCrossRefGoogle Scholar
  69. 69.
    Ledrappier F (1986) Dimension of invariant measures. Proceedings of the conference on ergodic theory and related topics II (Georgenthal, 1986). Teubner–texte Math 94:137–173Google Scholar
  70. 70.
    Ledrappier F, Misiurewicz M (1985) Dimension of Invariant Measures for Maps with Exponent Zero. Ergod Th Dynam Syst 5:595–610MathSciNetMATHCrossRefGoogle Scholar
  71. 71.
    Ledrappier F, Strelcyn JM (1982) A proof of the estimate from below in Pesin's entropy formula. Ergod Theory Dynam Syst 2:203–219MathSciNetMATHCrossRefGoogle Scholar
  72. 72.
    Ledrappier F, Young LS (1985) The Metric Entropy Of Diffeomorphisms. I Characterization Of Measures Satisfying Pesin's Entropy Formula. Ann Math 122:509–539MathSciNetMATHCrossRefGoogle Scholar
  73. 73.
    Ledrappier F, Young LS (1985) The Metric Entropy Of Diffeomorphisms, II. Relations Between Entropy, Exponents and Dimension. Ann Math 122:540–574MathSciNetCrossRefGoogle Scholar
  74. 74.
    Lopes A (1989) The Dimension Spectrum of the Maximal Measure. SIAM J Math Analysis 20:1243–1254MathSciNetMATHCrossRefGoogle Scholar
  75. 75.
    Mauldin RD, Urbański M (1996) Dimensions and measures in infinite iterated function systems. Proc Lond Math Soc 73(1):105–154Google Scholar
  76. 76.
    Mauldin RD, Urbański M (2002) Fractal measures for parabolic IFS. Adv Math 168(2):225–253Google Scholar
  77. 77.
    Mauldin DR, Urbański M (2003) Graph directed Markov systems. Geometry and dynamics of limit sets Cambridge Tracts in Mathematics, 148. Cambridge University Press, CambridgeGoogle Scholar
  78. 78.
    Mauldin RD; Urbański M (2000) Parabolic iterated function systems. Ergod Theory Dynam Syst 20(5):1423–1447Google Scholar
  79. 79.
    McCluskey H, Manning A (1983) Hausdorff Dimension For Horseshoes. Erg Th Dyn Syst 3:251–260MathSciNetCrossRefGoogle Scholar
  80. 80.
    Moran P (1946) Additive Functions Of Intervals and Hausdorff Dimension. Proceedings Of Cambridge Philosophical Society 42:15–23Google Scholar
  81. 81.
    Moreira C, Yoccoz J (2001) Stable Intersections of Cantor Sets with Large Hausdorff Dimension. Ann of Math (2) 154(1):45–96MathSciNetCrossRefGoogle Scholar
  82. 82.
    Mãné R (1981) On the Dimension of Compact Invariant Sets for Certain Nonlinear Maps. Lecture Notes in Mathematics, vol 898. SpringerGoogle Scholar
  83. 83.
    Mãné R (1990) The Hausdorff Dimension of Horseshoes of Surfaces. Bol Soc Bras Math 20:1–24Google Scholar
  84. 84.
    Neunhäuserer J (1999) An analysis of dimensional theoretical properties of some affine dynamical systems. Thesis. Free University Berlin, BerlinGoogle Scholar
  85. 85.
    Oseledets V (1968) A multiplicative ergodic theorem. Liapunov characteristic numbers for dynamical systems. Trans Moscow Math Soc 19:197–221MATHGoogle Scholar
  86. 86.
    Ott E, Sauer T, Yorke J (1994) Part I Background. In: Coping with chaos. Wiley Ser Nonlinear Sci, Wiley, New York, pp 1–62Google Scholar
  87. 87.
    Palis J, Takens F (1987) Hyperbolicity And The Creation Of Homoclinic Orbits. Ann Math 125:337–374MathSciNetMATHCrossRefGoogle Scholar
  88. 88.
    Palis J, Takens F (1993) Hyperbolicity And Sensitive Chaotic Dynamics At Homoclinic Bifurcations. Cambridge U Press, CambridgeMATHGoogle Scholar
  89. 89.
    Palis J, Takens F (1994) Homoclinic Tangencies For Hyperbolic Sets Of Large Hausdorff Dimension. Acta Math 172:91–136MathSciNetMATHCrossRefGoogle Scholar
  90. 90.
    Palis J, Viana M (1988) On the continuity of Hausdorff dimension and limit capacity for horseshoes. Lecture Notes in Math, vol 1331. SpringerGoogle Scholar
  91. 91.
    Pesin Y (1977) Characteristic exponents and smooth ergodic theory. Russian Math Surveys 32(4):55–114MathSciNetCrossRefGoogle Scholar
  92. 92.
    Pesin Y (1992) Dynamical systems with generalized hyperbolic attractors: hyperbolic, ergodic and topological properties. Erg Th Dyn Syst 12:123–151MathSciNetMATHCrossRefGoogle Scholar
  93. 93.
    Pesin Y (1993) On Rigorous Mathematical Definition of Correlation Dimension and Generalized Spectrum for Dimensions. J Statist Phys 71(3–4):529–547MathSciNetMATHCrossRefGoogle Scholar
  94. 94.
    Pesin Y (1997) Dimension Theory In Dynamical Systems: Rigorous Results And Applications. Cambridge U Press, CambridgeGoogle Scholar
  95. 95.
    Pesin Y (1997) Dimension theory in dynamical systems: contemporary views and applications. In: Chicago Lectures in Mathematics. Chicago University Press, ChicagoGoogle Scholar
  96. 96.
    Pesin Y, Pitskel' B (1984) Topological pressure and the variational principle for noncompact sets. Funct Anal Appl 18:307–318MathSciNetMATHCrossRefGoogle Scholar
  97. 97.
    Pesin Y, Sadovskaya V (2001) Multifractal analysis of conformal axiom A flows. Comm Math Phys 216(2):277–312MathSciNetMATHCrossRefGoogle Scholar
  98. 98.
    Pesin Y, Tempelman A (1995) Correlation Dimension of Measures Invariant Under Group Actions. Random Comput Dyn 3(3):137–156MathSciNetMATHGoogle Scholar
  99. 99.
    Pesin Y, Weiss H (1994) On the Dimension of Deterministic and Random Cantor‐like Sets. Math Res Lett 1:519–529MathSciNetMATHGoogle Scholar
  100. 100.
    Pesin Y, Weiss H (1996) On The Dimension Of Deterministic and Random Cantor‐like Sets, Symbolic Dynamics, And The Eckmann‐Ruelle Conjecture. Comm Math Phys 182:105–153MathSciNetMATHCrossRefGoogle Scholar
  101. 101.
    Pesin Y, Weiss H (1997) A Multifractal Analysis of Equilibrium Measures For Conformal Expanding Maps and Moran-like Geometric Constructions. J Stat Phys 86:233–275MathSciNetMATHCrossRefGoogle Scholar
  102. 102.
    Pesin Y, Weiss H (1997) The Multifractal Analysis of Gibbs Measures: Motivation. Mathematical Foundation and Examples. Chaos 7:89–106MathSciNetMATHCrossRefGoogle Scholar
  103. 103.
    Petersen K (1983) Ergodic theory. Cambridge studies in advanced mathematics 2. Cambridge Univ Press, CambridgeCrossRefGoogle Scholar
  104. 104.
    Pollicott M, Weiss H (1994) The Dimensions Of Some Self Affine Limit Sets In The Plane And Hyperbolic Sets. J Stat Phys 77:841–866MathSciNetMATHCrossRefGoogle Scholar
  105. 105.
    Pollicott M, Weiss H (1999) Multifractal Analysis for the Continued Fraction and Manneville‐Pomeau Transformations and Applications to Diophantine Approximation. Comm Math Phys 207(1):145–171MathSciNetMATHCrossRefGoogle Scholar
  106. 106.
    Przytycki F, Urbański M (1989) On Hausdorff Dimension of Some Fractal Sets. Studia Math 93:155–167Google Scholar
  107. 107.
    Ruelle D (1978) Thermodynamic Formalism. Addison‐WesleyGoogle Scholar
  108. 108.
    Ruelle D (1982) Repellers For Real Analytic Maps. Erg Th Dyn Syst 2:99–107MathSciNetMATHCrossRefGoogle Scholar
  109. 109.
    Schmeling J (1994) Hölder Continuity of the Holonomy Maps for Hyperbolic basic Sets II. Math Nachr 170:211–225MathSciNetMATHCrossRefGoogle Scholar
  110. 110.
    Schmeling J (1997) Symbolic Dynamics for Beta‐shifts and Self‐Normal Numbers. Erg Th Dyn Syst 17:675–694MathSciNetMATHCrossRefGoogle Scholar
  111. 111.
    Schmeling J (1998) A dimension formula for endomorphisms – the Belykh family. Erg Th Dyn Syst 18:1283–1309MathSciNetMATHCrossRefGoogle Scholar
  112. 112.
    Schmeling J (1999) On the Completeness of Multifractal Spectra. Erg Th Dyn Syst 19:1–22MathSciNetCrossRefGoogle Scholar
  113. 113.
    Schmeling J (2001) Entropy Preservation under Markov Codings. J Stat Phys 104(3–4):799–815MathSciNetMATHCrossRefGoogle Scholar
  114. 114.
    Schmeling J, Troubetzkoy S (1998) Dimension and invertibility of hyperbolic endomorphisms with singularities. Erg Th Dyn Syst 18:1257–1282MathSciNetMATHCrossRefGoogle Scholar
  115. 115.
    Schmeling J, Weiss H (2001) Dimension theory and dynamics. AMS Proceedings of Symposia in Pure Mathematics series 69:429–488Google Scholar
  116. 116.
    Series C (1985) The Modular Surface and Continued Fractions. J Lond Math Soc 31:69–80MathSciNetMATHCrossRefGoogle Scholar
  117. 117.
    Simon K (1997) The Hausdorff dimension of the general Smale–Williams solenoidwith different contraction coefficients. Proc Am Math Soc 125:1221–1228MATHCrossRefGoogle Scholar
  118. 118.
    Simpelaere D (1994) Dimension Spectrum of Axiom-A Diffeomorphisms, II. Gibbs Measures. J Stat Phys 76:1359–1375MathSciNetMATHCrossRefGoogle Scholar
  119. 119.
    Solomyak B (1995) On the random series \({\sum\pm\lambda\sp n}\) (an Erdös problem). Ann of Math (2) 142(3):611–625MathSciNetCrossRefGoogle Scholar
  120. 120.
    Solomyak B (2004) Notes on Bernoulli convolutions. In: Fractal geometry and applications: a jubilee of Benoît Mandelbrot. Proc Sympos Pure Math, 72, Part 1. Amer Math Soc, Providence, pp 207–230CrossRefGoogle Scholar
  121. 121.
    Stratmann B (1995) Fractal Dimensions for Jarnik Limit Sets of Geometrically Finite Kleinian Groups; The Semi‐Classical Approach. Ark Mat 33:385–403MathSciNetMATHCrossRefGoogle Scholar
  122. 122.
    Takens F (1981) Detecting Strange Attractors in Turbulence. Lecture Notes in Mathematics, vol 898. SpringerGoogle Scholar
  123. 123.
    Takens F, Verbitzki E (1999) Multifractal analysis of local entropies for expansive homeomorphisms with specification. Comm Math Phys 203:593–612MathSciNetMATHCrossRefGoogle Scholar
  124. 124.
    Walters P (1982) Introduction to Ergodic Theory. SpringerGoogle Scholar
  125. 125.
    Weiss H (1992) Some Variational Formulas for Hausdorff Dimension, Topological Entropy, and SRB Entropy for Hyperbolic Dynamical System. J Stat Phys 69:879–886MATHCrossRefGoogle Scholar
  126. 126.
    Weiss H (1999) The Lyapunov Spectrum Of Equilibrium Measures for Conformal Expanding Maps and Axiom-A Surface Diffeomorphisms. J Statist Phys 95(3–4):615–632MathSciNetMATHCrossRefGoogle Scholar
  127. 127.
    Young LS (1981) Capacity of Attractors. Erg Th Dyn Syst 1:381–388MATHCrossRefGoogle Scholar
  128. 128.
    Young LS (1982) Dimension, Entropy, and Lyapunov Exponents. Erg Th Dyn Syt 2:109–124MATHCrossRefGoogle Scholar

Books and Reviews

  1. 129.
    Bowen R (1975) Equilibrium states and the ergodic theory of Anosov diffeomorphisms. Lecture Notes in Mathematics, vol 470. SpringerGoogle Scholar
  2. 130.
    Eckmann JP, Ruelle D (1985) Ergodic Theory Of Chaos And Strange Attractors. Rev Mod Phys 57:617–656MathSciNetCrossRefGoogle Scholar
  3. 131.
    Federer H (1969) Geometric measure theory. SpringerGoogle Scholar
  4. 132.
    Hasselblatt B, Katok A (2002) Handbook of Dynamical Systems, vol 1, Survey 1. Principal Structures. ElsevierGoogle Scholar
  5. 133.
    Katok A (1980) Lyapunov exponents, entropy and periodic orbits for diffeomorphisms. Inst Hautes Études Sci Publ Math 51:137–173MathSciNetMATHCrossRefGoogle Scholar
  6. 134.
    Katok A, Hasselblatt B (1995) Introduction to the Modern Theory of Dynamical Systems. Cambridge Univ Press, CambridgeMATHCrossRefGoogle Scholar
  7. 135.
    Keller G (1998) Equilibrium states in ergodic theory. In: London Mathematical Society Student Texts 42. Cambridge University Press, CambridgeGoogle Scholar
  8. 136.
    Mañé R (1987) Ergodic theory and differentiable dynamics. In: Ergebnisse der Mathematik und ihrer Grenzgebiete 3, Band 8. SpringerGoogle Scholar
  9. 137.
    Mario R, Urbański M (2005) Regularity properties of Hausdorff dimension in infinite conformal iterated function systems. Ergod Theory Dynam Syst 25(6):1961–1983Google Scholar
  10. 138.
    Mattila P (1995) Geometry of sets and measures in Euclidean spaces. In: Fractals and rectifiability. Cambridge University Press, CambridgeGoogle Scholar
  11. 139.
    Pugh C, Shub M (1989) Ergodic attractors. Trans Amer Math Soc 312(1):1–54MathSciNetMATHCrossRefGoogle Scholar
  12. 140.
    Takens F (1988) Limit capacity and Hausdorff dimension of dynamically defined Cantor sets. Lecture Notes in Math, vol 1331. SpringerGoogle Scholar

Copyright information

© Springer-Verlag 2012

Authors and Affiliations

  • Jörg Schmeling
    • 1
  1. 1.Center for Mathematical SciencesLund UniversityLundSweden