Hostname: page-component-7dd5485656-tbj44 Total loading time: 0 Render date: 2025-10-30T19:41:12.535Z Has data issue: false hasContentIssue false
  • English
  • Français

Dimension estimates in smooth dynamics: a survey of recent results

Published online by Cambridge University Press:  26 May 2010

LUIS BARREIRA  and
KATRIN GELFERT
LUIS BARREIRA
Affiliation:
Departamento de Matemática, Instituto Superior Técnico, 1049-001 Lisboa, Portugal (email: luis.barreira@math.ist.utl.pt)
KATRIN GELFERT
Affiliation:
IMPA, Estrada Dona Castorina 110, Rio de Janeiro/Brasil 22460-320, Brazil (email: katrin.gelfert@googlemail.com)
Get access Rights & Permissions [Opens in a new window]

Abstract

We survey a collection of results in the dimension theory of dynamical systems, with emphasis on the study of repellers and hyperbolic sets of smooth dynamics. We discuss the most preeminent results in the area as well as the main difficulties in developing a general theory. Despite many interesting and non-trivial developments, only the case of conformal dynamics is completely understood. The study of the dimension of invariant sets of non-conformal maps has unveiled several new phenomena, but it still lacks today a satisfactory general approach. Indeed, we have a complete understanding of only a few classes of invariant sets of non-conformal maps satisfying certain simplifying assumptions. For example, the assumptions may ensure that there is a clear separation between different Lyapunov directions or that number-theoretical properties do not influence the dimension.

Information

Type
SURVEY
Information
Ergodic Theory and Dynamical Systems , Volume 31 , Issue 3 , June 2011 , pp. 641 - 671
Copyright
Copyright © Cambridge University Press 2010

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

[1]Babin, A. and Vishik, M.. Attractors of Evolution Equations (Studies in Mathematics and its Applications, 25). North-Holland, Amsterdam, 1992.Google Scholar
[2]Barreira, L.. A non-additive thermodynamic formalism and applications to dimension theory of hyperbolic dynamical systems. Ergod. Th. & Dynam. Sys. 16 (1996), 871–927.CrossRefGoogle Scholar
[3]Barreira, L.. Dimension estimates in nonconformal hyperbolic dynamics. Nonlinearity 16 (2003), 1657–1672.CrossRefGoogle Scholar
[4]Barreira, L.. Dimension and Recurrence in Hyperbolic Dynamics (Progress in Mathematics, 272). Birkhäuser, Basel, 2008.Google Scholar
[5]Barreira, L.. Almost additive thermodynamic formalism: some recent developments. Preprint.Google Scholar
[6]Barreira, L. and Gelfert, K.. Multifractal analysis for Lyapunov exponents on nonconformal repellers. Comm. Math. Phys. 267 (2006), 393–418.CrossRefGoogle Scholar
[7]Barreira, L. and Wolf, C.. Measures of maximal dimension for hyperbolic diffeomorphisms. Comm. Math. Phys. 239 (2003), 93–113.CrossRefGoogle Scholar
[8]Bedford, T.. Crinkly curves, Markov partitions and box dimension of self-similar sets. PhD Thesis, University of Warwick, 1984.Google Scholar
[9]Bedford, T.. The box dimension of self-affine graphs and repellers. Nonlinearity 2 (1989), 53–71.CrossRefGoogle Scholar
[10]Bedford, T. and Urbański, M.. The box and Hausdorff dimension of self-affine sets. Ergod. Th. & Dynam. Sys. 10 (1990), 627–644.CrossRefGoogle Scholar
[11]Ben-Artzi, A., Eden, A., Foias, C. and Nicolaenko, B.. Hölder continuity for the inverse of Mañé’s projection. J. Math. Anal. Appl. 178 (1993), 22–29.CrossRefGoogle Scholar
[12]Blinchevskaya, M. and Ilyashenko, Yu.. Estimates for the entropy dimension of the maximal attractor for k-contracting systems in an infinite-dimensional space. Russ. J. Math. Phys. 6 (1999), 20–26.Google Scholar
[13]Bonatti, C., Díaz, L. and Viana, M.. Discontinuity of the Hausdorff dimension of hyperbolic sets. C. R. Acad. Sci. Paris Sér. I Math. 320 (1995), 713–718.Google Scholar
[14]Bothe, H.. The Hausdorff dimension of certain solenoids. Ergod. Th. & Dynam. Sys. 15 (1995), 449–474.CrossRefGoogle Scholar
[15]Bowen, R.. Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms (Lecture Notes in Mathematics, 470). Springer, Berlin–New York, 1975.CrossRefGoogle Scholar
[16]Bowen, R.. Hausdorff dimension of quasi-circles. Inst. Hautes Études Sci. Publ. Math. 50 (1979), 259–273.CrossRefGoogle Scholar
[17]Chepyzhov, V. and Ilyin, A.. A note on the fractal dimension of attractors of dissipative dynamical systems. Nonlinear Anal. 44A (2001), 811–819.CrossRefGoogle Scholar
[18]Díaz, L. and Viana, M.. Discontinuity of Hausdorff dimension and limit capacity on arcs of diffeomorphisms. Ergod. Th. & Dynam. Sys. 9 (1989), 403–425.CrossRefGoogle Scholar
[19]Douady, A. and Oesterlé, J.. Dimension de Hausdorff des attracteurs. C. R. Acad. Sci. Paris 290 (1980), 1135–1138.Google Scholar
[20]Dysman, M.. Fractal dimension for repellers of maps with holes. J. Stat. Phys. 120 (2005), 479–509.CrossRefGoogle Scholar
[21]Eden, A.. Local estimates for the Hausdorff dimension of an attractor. J. Math. Anal. Appl. 150 (1990), 100–119.CrossRefGoogle Scholar
[22]Eden, A., Foias, C. and Temam, R.. Local and global Lyapunov exponents. J. Dynam. Differential Equations 3 (1991), 133–177.CrossRefGoogle Scholar
[23]Eden, A., Foias, C. and Nicolaenko, B.. Exponential attractors of optimal Lyapunov dimension for Navier–Stokes equations. C. R. Acad. Sci. Paris Sér. I Math. 316 (1993), 1211–1215.Google Scholar
[24]Eden, A., Foias, C., Nicolaenko, B. and Temam, R.. Exponential Attractors for Dissipative Evolution Equations (Research in Applied Mathematics, 37). Wiley, New York, 1994.Google Scholar
[25]Falconer, K.. The Hausdorff dimension of self-affine fractals. Math. Proc. Cambridge Philos. Soc. 103 (1988), 339–350.CrossRefGoogle Scholar
[26]Falconer, K.. A subadditive thermodynamic formalism for mixing repellers. J. Phys. A 21 (1988), L737–L742.CrossRefGoogle Scholar
[27]Falconer, K.. Dimensions and measures of quasi self-similar sets. Proc. Amer. Math. Soc. 106 (1989), 543–554.CrossRefGoogle Scholar
[28]Falconer, K.. The dimension of self-affine fractals. II. Math. Proc. Cambridge Philos. Soc. 111 (1992), 169–179.CrossRefGoogle Scholar
[29]Falconer, K.. Bounded distortion and dimension for non-conformal repellers. Math. Proc. Cambridge Philos. Soc. 115 (1994), 315–334.CrossRefGoogle Scholar
[30]Falconer, K.. Fractal Geometry (Mathematical Foundations and Applications). Wiley, Hoboken, NJ, 2003.CrossRefGoogle Scholar
[31]Fan, A., Jiang, Y. and Wu, J.. Asymptotic Hausdorff dimensions of Cantor sets associated with an asymptotically non-hyperbolic family. Ergod. Th. & Dynam. Sys. 25 (2005), 1799–1808.CrossRefGoogle Scholar
[32]Fathi, A.. Expansiveness, hyperbolicity, and Hausdorff dimension. Comm. Math. Phys. 126 (1989), 249–262.CrossRefGoogle Scholar
[33]Foias, C. and Olson, E.. Finite fractal dimension and Hölder–Lipschitz parametrization. Indiana Univ. Math. J. 45 (1996), 603–616.CrossRefGoogle Scholar
[34]Franz, A.. Hausdorff dimension estimates for invariant sets with an equivariant tangent bundle splitting. Nonlinearity 11 (1998), 1063–1074.CrossRefGoogle Scholar
[35]Gatzouras, D. and Lalley, S.. Hausdorff and box dimensions of certain self-affine fractals. Indiana Univ. Math. J. 41 (1992), 533–568.Google Scholar
[36]Gatzouras, D. and Peres, Y.. Invariant measures of full dimension for some expanding maps. Ergod. Th. & Dynam. Sys. 17 (1997), 147–167.CrossRefGoogle Scholar
[37]Gelfert, K.. Maximal local Lyapunov dimension bounds the box dimension. Direct proof for invariant sets on Riemannian manifolds. Z. Anal. Anwend. 22 (2003), 553–568.CrossRefGoogle Scholar
[38]Gelfert, K.. Dimension estimates beyond conformal and hyperbolic dynamics. Dyn. Syst. 20 (2005), 267–280.CrossRefGoogle Scholar
[39]Gu, X.. An upper bound for the Hausdorff dimension of a hyperbolic set. Nonlinearity 4 (1991), 927–934.CrossRefGoogle Scholar
[40]Hale, J.. Asymptotic Behavior of Dissipative Systems (Mathematical Surveys and Monographs, 25). American Mathematical Society, Providence, RI, 1988.Google Scholar
[41]Hasselblatt, B.. Regularity of the Anosov splitting and of horospheric foliations. Ergod. Th. & Dynam. Sys. 14 (1994), 645–666.CrossRefGoogle Scholar
[42]Hasselblatt, B. and Schmeling, J.. Dimension product structure of hyperbolic sets. Electron. Res. Announc. Amer. Math. Soc. 10 (2004), 88–96.CrossRefGoogle Scholar
[43]Hasselblatt, B. and Schmeling, J.. Dimension product structure of hyperbolic sets. Modern Dynamical Systems and Applications. Cambridge University Press, Cambridge, 2004, pp. 331–345.Google Scholar
[44]Hirayama, M.. An upper estimate of the Hausdorff dimension of stable sets. Ergod. Th. & Dynam. Sys. 24 (2004), 1109–1125.CrossRefGoogle Scholar
[45]Horita, V. and Viana, M.. Hausdorff dimension of non-hyperbolic repellers. I. Maps with holes. J. Statist. Phys. 105 (2001), 835–862.CrossRefGoogle Scholar
[46]Horita, V. and Viana, M.. Hausdorff dimension for non-hyperbolic repellers. II: DA diffeomorphisms. Discrete Contin. Dyn. Syst. 13 (2005), 1125–1152.CrossRefGoogle Scholar
[47]Hu, H.. Box dimensions and topological pressure for some expanding maps. Comm. Math. Phys. 191 (1998), 397–407.CrossRefGoogle Scholar
[48]Hueter, I. and Lalley, S.. Falconer’s formula for the Hausdorff dimension of a self-affine set in ℝ2. Ergod. Th. & Dynam. Sys. 15 (1995), 77–97.CrossRefGoogle Scholar
[49]Hunt, B.. Maximum local Lyapunov dimension bounds the box dimension of chaotic attractors. Nonlinearity 9 (1996), 845–852.CrossRefGoogle Scholar
[50]Hunt, B. and Kaloshin, V.. Regularity of embeddings of infinite-dimensional fractal sets into finite-dimensional spaces. Nonlinearity 12 (1999), 1263–1275.CrossRefGoogle Scholar
[51]Hunt, B., Sauer, T. and Yorke, J.. Prevalence: a translation-invariant “almost every” on infinite-dimensional spaces. Bull. Amer. Math. Soc. (N.S.) 27 (1992), 217–238.CrossRefGoogle Scholar
[52]Hutchinson, J.. Fractals and self-similarity. Indiana Univ. Math. J. 30 (1981), 713–747.CrossRefGoogle Scholar
[53]Kaplan, J. and Yorke, J.. Chaotic behavior of multidimensional difference equations. Functional Differential Equations and Approximation of Fixed Points, Proc. Summer School and Conf. (Bonn, 1978) (Lecture Notes in Mathematics, 730). Eds. Peitgen, H.-O. and Walther, H.-O.. Springer, Berlin, 1979, pp. 204–227.CrossRefGoogle Scholar
[54]Kenyon, R. and Peres, Y.. Measures of full dimension on affine-invariant sets. Ergod. Th. & Dynam. Sys. 16 (1996), 307–323.CrossRefGoogle Scholar
[55]Ledrappier, F.. Some relations between dimension and Lyapunov exponents. Comm. Math. Phys. 81 (1981), 229–238.CrossRefGoogle Scholar
[56]Ledrappier, F. and Young, L.-S.. The metric entropy of diffeomorphisms. II: Relations between entropy, exponents and dimension. Ann. of Math. (2) 122 (1985), 540–574.CrossRefGoogle Scholar
[57]Leonov, G.. On estimates of the Hausdorff dimension of attractors. Vestn. Leningr. Univ., Ser. I 3 (1991), 41–44.Google Scholar
[58]Leonov, G.. Lyapunov dimensions formulas for Hénon and Lorenz attractors. Algebra i Analiz 13 (2001), 453–464.Google Scholar
[59]Luzia, N.. Hausdorff dimension for an open class of repellers in ℝ2. Nonlinearity 19 (2006), 2895–2908.CrossRefGoogle Scholar
[60]Luzia, N. A variational principle for the dimension for a class of non-conformal repellers. Ergod. Th. & Dynam. Sys. 26 (2006), 821–845.CrossRefGoogle Scholar
[61]Luzia, N.. Measure of full dimension for some nonconformal repellers. Discrete Contin. Dyn. Syst. 26 (2010), 291–302.CrossRefGoogle Scholar
[62]Mallet-Paret, J.. Negatively invariant sets of compact maps and an extension of a theorem of Cartwright. J. Differential Equations 22 (1976), 331–348.CrossRefGoogle Scholar
[63]Mañé, R.. On the dimension of the compact invariant sets of certain non-linear maps. Dynamical Systems and Turbulence (Warwick, 1979/80) (Lecture Notes in Mathematics, 898). Springer, Berlin–New York, 1981, pp. 230–242.CrossRefGoogle Scholar
[64]Mañé, R.. The Hausdorff dimension of horseshoes of diffeomorphisms of surfaces. Bol. Soc. Brasil. Mat. (N.S.) 20 (1990), 1–24.CrossRefGoogle Scholar
[65]Manning, A. and Simon, K.. Subadditive pressure for triangular maps. Nonlinearity 20 (2007), 133–149.CrossRefGoogle Scholar
[66]McCluskey, H. and Manning, A.. Hausdorff dimension for horseshoes. Ergod. Th. & Dynam. Sys. 3 (1983), 251–260.CrossRefGoogle Scholar
[67]McMullen, C.. The Hausdorff dimension of general Sierpiński carpets. Nagoya Math. J. 96 (1984), 1–9.CrossRefGoogle Scholar
[68]Moran, P.. Additive functions of intervals and Hausdorff measure. Proc. Cambridge Philos. Soc. 42 (1946), 15–23.CrossRefGoogle Scholar
[69]Neunhäuserer, J.. Number theoretical peculiarities in the dimension theory of dynamical systems. Israel J. Math. 128 (2002), 267–283.CrossRefGoogle Scholar
[70]Noack, A. and Reitmann, V.. Hausdorff dimension estimates for invariant sets of time-dependent vector fields. Z. Anal. Anwend. 15 (1996), 457–473.CrossRefGoogle Scholar
[71]Palis, J. and Takens, F.. Hyperbolicity and Sensitive Chaotic Dynamics at Homoclinic Bifurcations. Cambridge University Press, Cambridge, 1993.Google Scholar
[72]Palis, J. and Viana, M.. On the continuity of Hausdorff dimension and limit capacity for horseshoes. Dynamical Systems (Valparaiso, 1986) (Lecture Notes in Mathematics, 1331). Eds. Bamón, R., Labarca, R. and Palis, J.. Springer, Berlin, 1988, pp. 150–160.CrossRefGoogle Scholar
[73]Peres, Y.. The self-affine carpets of McMullen and Bedford have infinite Hausdorff measure. Math. Proc. Cambridge Philos. Soc. 116 (1994), 513–526.CrossRefGoogle Scholar
[74]Pesin, Ya.. Dimension Theory in Dynamical Systems: Contemporary Views and Applications (Chicago Lectures in Mathematics). Chicago University Press, Chicago, 1997.CrossRefGoogle Scholar
[75]Pollicott, M. and Weiss, H.. The dimensions of some self-affine limit sets in the plane and hyperbolic sets. J. Statist. Phys. 77 (1994), 841–866.CrossRefGoogle Scholar
[76]Przytycki, F. and Urbański, M.. On the Hausdorff dimension of some fractal sets. Studia Math. 93 (1989), 155–186.CrossRefGoogle Scholar
[77]Rams, M.. Measures of maximal dimension for linear horseshoes. Real Anal. Exchange 31 (2005/06), 55–62.CrossRefGoogle Scholar
[78]Ruelle, D.. Statistical mechanics on a compact set with ℤν action satisfying expansiveness and specification. Trans. Amer. Math. Soc. 185 (1973), 237–251.CrossRefGoogle Scholar
[79]Ruelle, D.. Thermodynamic Formalism (Encyclopedia of Mathematics and its Applications, 5). Addison-Wesley, Reading, MA, 1978.Google Scholar
[80]Ruelle, D.. Repellers for real analytic maps. Ergod. Th. & Dynam. Sys. 2 (1982), 99–107.CrossRefGoogle Scholar
[81]Schmeling, J. and Weiss, H.. An overview of the dimension theory of dynamical systems. Smooth Ergodic Theory and its Applications (Seattle, 1999) (Proceedings of Symposia in Pure Mathematics, 69). Eds. Katok, A., de la Llave, R., Pesin, Ya. and Weiss, H.. American Mathematical Society, Providence, RI, 2001, pp. 429–488.CrossRefGoogle Scholar
[82]Shafikov, R. and Wolf, C.. Stable sets, hyperbolicity and dimension. Discrete Contin. Dyn. Syst. 12 (2005), 403–412.CrossRefGoogle Scholar
[83]Simon, K.. Hausdorff dimension for noninvertible maps. Ergod. Th. & Dynam. Sys. 13 (1993), 199–212.CrossRefGoogle Scholar
[84]Simon, K.. The Hausdorff dimension of the Smale–Williams solenoid with different contraction coefficients. Proc. Amer. Math. Soc. 125 (1997), 1221–1228.CrossRefGoogle Scholar
[85]Simon, K. and Solomyak, B.. Hausdorff dimension for horseshoes in ℝ3. Ergod. Th. & Dynam. Sys. 19 (1999), 1343–1363.CrossRefGoogle Scholar
[86]Solomyak, B.. Measure and dimension for some fractal families. Math. Proc. Cambridge Philos. Soc. 124 (1998), 531–546.CrossRefGoogle Scholar
[87]Takens, F. and Palis, . Limit capacity and Hausdorff dimension of dynamically defined Cantor sets. Dynamical Systems (Valparaiso, 1986) (Lecture Notes in Mathematics, 1331). Eds. Bamón, R. and Labarca, R.. Springer, Berlin, 1988, pp. 196–212.CrossRefGoogle Scholar
[88]Temam, R.. Infinite-Dimensional Dynamical Systems in Mechanics and Physics (Applied Mathematical Sciences, 68). Springer, New York, 1988.CrossRefGoogle Scholar
[89]Urbański, M.. Measures and dimensions in conformal dynamics. Bull. Amer. Math. Soc. 40 (2003), 281–321.CrossRefGoogle Scholar
[90]Urbański, M. and Wolf, C.. SRB measures for Axiom A endomorphisms. Math. Res. Lett. 11 (2004), 785–797.CrossRefGoogle Scholar
[91]Urbański, M. and Wolf, C.. Ergodic theory of parabolic horseshoes. Comm. Math. Phys. 281 (2008), 711–751.CrossRefGoogle Scholar
[92]Walters, P.. A variational principle for the pressure of continuous transformations. Amer. J. Math. 97 (1976), 937–971.CrossRefGoogle Scholar
[93]Walters, P.. An Introduction to Ergodic Theory (Graduate Texts in Mathematics, 79). Springer, Berlin, 1981.Google Scholar
[94]Wolf, C.. On measures of maximal and full dimension for polynomial automorphisms of ℂ2. Trans. Amer. Math. Soc. 355 (2003), 3227–3239.CrossRefGoogle Scholar
[95]Yayama, Y.. Dimensions of compact invariant sets of some expanding maps. Ergod. Th. & Dynam. Sys. 29 (2009), 281–315.CrossRefGoogle Scholar
[96]Young, L.-S.. Dimension, entropy and Lyapunov exponents. Ergod. Th. & Dynam. Sys. 2 (1982), 109–124.CrossRefGoogle Scholar
[97]Zhang, Y.. Dynamical upper bounds for Hausdorff dimension of invariant sets. Ergod. Th. & Dynam. Sys. 17 (1997), 739–756.CrossRefGoogle Scholar