Dimension product structure of hyperbolic sets
Authors:
Boris Hasselblatt and Jörg Schmeling
Journal:
Electron. Res. Announc. Amer. Math. Soc. 10 (2004), 88-96
MSC (2000):
Primary 37D10; Secondary 37C35
DOI:
https://doi.org/10.1090/S1079-6762-04-00133-7
Published electronically:
August 26, 2004
MathSciNet review:
2084468
Full-text PDF Free Access
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Abstract: We conjecture that the fractal dimension of hyperbolic sets can be computed by adding those of their stable and unstable slices. This would facilitate substantial progress in the calculation or estimation of these dimensions, which are related in deep ways to dynamical properties. We prove the conjecture in a model case of Smale solenoids.
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[A]Anosov Dmitriĭ Anosov: Geodesic flows on Riemann manifolds with negative curvature, Proc. Steklov Inst. 90 (1967).
[AS]AnosovSinai Dmitriĭ Anosov, Yakov Sinai: Some smooth ergodic systems, Russian Math. Surveys 22 (1967), no. 5, 103–167.
[BS]BS L. Barreira and J. Schmeling: Sets of “non-typical” points have full topological entropy and full Hausdorff dimension, Israel Journal of Mathematics 116 (2000), 29–70.
[BPS1]BarreiraPesinSchmeling Luís Barreira, Yakov Pesin, Jörg Schmeling: On the pointwise dimension of hyperbolic measures: a proof of the Eckmann-Ruelle conjecture, Electronic Research Announcements of the AMS 2 (1996), no. 1, 69–72 (electronic); Dimension and product structure of hyperbolic measures, Annals of Mathematics (2) 149 (1999), no. 3, 755–783. ;
[B]B Hans-Günter Bothe: The dimension of some solenoids, Ergodic Theory and Dynamical Systems 15 (1995), no. 3, 449–474
[Bow]Bowen Rufus Bowen, Topological entropy for noncompact sets, Transactions of the AMS 184 (1973), 125–136.
[BS]BrinStuck Michael Brin, Garrett Stuck: Introduction to dynamical systems. Cambridge University Press, Cambridge, 2002.
[F]Falconer Kenneth Falconer: The geometry of fractal sets. Cambridge University Press, Cambridge, 1986.
[H1]thesis Boris Hasselblatt: Regularity of the Anosov splitting and of horospheric foliations, Ergodic Theory and Dynamical Systems 14 (1994), no. 4, 645–666.
[H2]Hasselblattsurvey Boris Hasselblatt: Hyperbolic dynamics, in Handbook of Dynamical Systems 1A, Elsevier, 2002.
[HS]HasselblattSchmeling Boris Hasselblatt, Jörg Schmeling: Dimension product structure of hyperbolic sets. In Modern Dynamical Systems and Applications, B. Hasselblatt, M. Brin, Y. Pesin, eds., Cambridge University Press, New York, 2004, pp. 331–345.
[HW]HasselblattWilkinson Boris Hasselblatt, Amie Wilkinson: Prevalence of non-Lipschitz Anosov foliations, Ergodic Theory and Dynamical Systems 19 (1999), no. 3, 643–656; ERA-AMS 3 (1997), 93–98. ;
[Ho]Hopf Eberhard Hopf: Statistik der geodätischen Linien in Mannigfaltigkeiten negativer Krümmung, Berichte über die Verhandlungen der Sächsischen Akademie der Wissenschaften zu Leipzig, Mathematisch-Physikalische Klasse 91 (1939), 261–304.
[KH]KatokHasselblatt Anatole Katok, Boris Hasselblatt: Introduction to the modern theory of dynamical systems, Encyclopedia of Mathematics and its Applications 54, Cambridge University Press, 1995.
[LY]LY1 François Ledrappier, Lai-Sang Young: The metric entropy of diffeomorphisms. I. Characterization of measures satisfying Pesin’s entropy formula; II. Relations between entropy, exponents and dimension, Annals of Mathematics (2) 122 (1985), no. 3, 509–539; 540–574. ;
[P]PesinBook Ya. Pesin, Dimension theory in dynamical systems: contemporary views and applications, Chicago Lectures in Mathematics, Chicago University Press, 1997.
[PSW]PughShubWilkinson Charles C. Pugh, Michael Shub, Amie Wilkinson: Hölder foliations, Duke Math. J. 86 (1997), no. 3, 517–546.
[S]Schmeling Jörg Schmeling: Hölder continuity of the holonomy maps for hyperbolic basic sets. II, Mathematische Nachrichten 170 (1994), 211–225.
[SS]SchmSieg Jörg Schmeling, Rainer Siegmund-Schultze: Hölder continuity of the holonomy maps for hyperbolic basic sets. I., Ergodic theory and related topics, III (Güstrow, 1990), pp. 174–191, Springer Lecture Notes in Mathematics 1514, Springer-Verlag, Berlin, 1992.
[Sm]Smale Steven Smale: Differentiable dynamical systems, Bulletin of the AMS 73 (1967), 747–817.
[Y]YoungSRB Lai-Sang Young: What are SRB measures, and which dynamical systems have them?, J. Statistical Physics 108 (2002), no. 5/6, 733–754.
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Additional Information
Boris Hasselblatt
Affiliation:
Department of Mathematics, Tufts University, Medford, MA 02155
MR Author ID:
270790
Email:
bhasselb@tufts.edu
Jörg Schmeling
Affiliation:
Lund Institute of Technology, Lunds Universitet, Box 118, SE-22100 Lund, Sweden
Email:
Jorg.Schmeling@math.lth.se
Keywords:
Hyperbolic set,
fractal dimension,
Hausdorff dimension,
Eckmann-Ruelle conjecture,
holonomies,
Lipschitz continuity,
product structure
Received by editor(s):
June 8, 2004
Published electronically:
August 26, 2004
Communicated by:
Svetlana Katok
Article copyright:
© Copyright 2004
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
