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Ahlfors regularity and fractal dimension of Smale spaces

Part of: Dynamical systems with hyperbolic behavior Topological dynamics

Published online by Cambridge University Press:  29 April 2021

DIMITRIS MICHAIL GERONTOGIANNIS
DIMITRIS MICHAIL GERONTOGIANNIS*
Affiliation:
School of Mathematics and Statistics, University of Glasgow, University Place, Glasgow, G12 8QQ, UK
*
e-mail: [email protected]
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Abstract

We prove that, up to topological conjugacy, every Smale space admits an Ahlfors regular Bowen measure. Bowen’s construction of Markov partitions implies that Smale spaces are factors of topological Markov chains. The latter are equipped with Parry’s measure, which is Ahlfors regular. By extending Bowen’s construction, we create a tool for transferring the Ahlfors regularity of the Parry measure down to the Bowen measure of the Smale space. An essential part of our method uses a refined notion of approximation graphs over compact metric spaces. Moreover, we obtain new estimates for the Hausdorff, box-counting and Assouad dimensions of a large class of Smale spaces.

Keywords

Ahlfors regularfractal dimensionhyperbolicityMarkov partition

MSC classification

Primary: 37D20: Uniformly hyperbolic systems (expanding, Anosov, Axiom A, etc.)
Secondary: 37B10: Symbolic dynamics
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 42 , Issue 7 , July 2022 , pp. 2281 - 2332
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Copyright
© The Author(s), 2021. Published by Cambridge University Press

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References

Adler, R. L.. Symbolic dynamics and Markov partitions. Bull. Amer. Math. Soc. 35 (1998), 1–56.CrossRefGoogle Scholar
Adler, R. L., Konheim, A. G. and McAndrew, M. H.. Topological entropy. Trans. Amer. Math. Soc. 114 (1965), 309–319.CrossRefGoogle Scholar
Ahlfors, L.. Zur Theorie der Uberlagerungsflachen. Acta Math. 65 (1935), 157–194.CrossRefGoogle Scholar
Ambrosio, L. and Tilli, P.. Topics on Analysis in Metric Spaces. Oxford University Press, Oxford, UK, 2004.Google Scholar
Amini, M., Elliott, G. A. and Golestani, N.. The category of Bratteli diagrams. Canad. J. Math. 67 (2015), 990–1023.CrossRefGoogle Scholar
Artigue, A.. Self-similar hyperbolicity. Ergod. Th. & Dynam. Sys. 38 (2018), 2422–2446.CrossRefGoogle Scholar
Assouad, P.. Étude d’une dimension métrique liée àl la possibilité de plongements dans ${\mathbb{R}}^n$ . C. R. Acad. Sci. Paris 288 (1979), 731–734.Google Scholar
Assouad, P.. Pseudodistances, facteurs et dimension métrique, Séminaire d’Analyse Harmonique 1979–1980. Publ. Math. Orsay 80 (1980), 1–33.Google Scholar
Assouad, P.. Plongements Lipschitziens dans ${\mathbb{R}}^n$ . Bull. Soc. Math. France 111 (1983), 429–448.CrossRefGoogle Scholar
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
Barreira, L.. Dimension and Recurrence in Hyperbolic Dynamics. Birkhäuser, Basel, CH, 2008.Google Scholar
Barreira, L.. Thermodynamic Formalism and Applications to Dimension Theory. Birkhäuser, Basel, CH, 2011.CrossRefGoogle Scholar
Bowen, R.. Markov partitions and minimal sets for Axiom A diffeomorphisms Amer. J. Math. 92 (1970), 907–918.CrossRefGoogle Scholar
Bowen, R.. Markov partitions for Axiom A diffeomorphisms. Amer. J. Math. 92 (1970), 725–747.CrossRefGoogle Scholar
Bowen, R.. Periodic points and measures for Axiom A diffeomorphisms. Trans. Amer. Math. Soc. 154 (1971), 377–397.Google Scholar
Bowen, R.. Hausdorff dimension of quasi-circles . Publ. Math. Inst. Hautes Études Sci. 50 (1979), 259–273.CrossRefGoogle Scholar
Bowen, R.. Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms . Springer, Berlin, DE, 2008.CrossRefGoogle Scholar
Brin, M. and Stuck, G.. Introduction to Dynamical Systems. Cambridge University Press, Cambridge, UK, 2002.CrossRefGoogle Scholar
Brodskiy, N., Dydak, J., Higes, J. and Mitra, A.. Assouad-Nagata dimension via Lipschitz extensions. Israel J. Math. 171 (2009), 405–423.CrossRefGoogle Scholar
Christensen, E. and Ivan, C.. Sums of two dimensional spectral triples. Math. Scand. 100 (2007), 35–60.CrossRefGoogle Scholar
Connes, A.. Noncommutative Geometry. Academic Press Inc, London, UK and San Diego, CA, 1994.Google Scholar
Coornaert, M.. Mesures de Patterson-Sullivan sure le bord d’un espace hyperbolique au sensde Gromov. Pacific. J. Math. 159 (1993), 241–270.CrossRefGoogle Scholar
Cuntz, J.. Simple ${C}^{\ast }$ -algebras generated by isometries. Comm. Math. Phys. 57 (1977), 173–185.CrossRefGoogle Scholar
Diestel, R.. Graph Theory. Springer, New York, NY, 2005.Google Scholar
Le Donne, E. and Rajala, T.. Assouad dimension, Nagata dimension, and uniformly close metric tangents. Indiana Univ. Math. J. 64 (2015), 21–54.CrossRefGoogle Scholar
Falconer, K. J.. Fractal Geometry: Mathematical Foundations and Applications. John Wiley & Sons, Hoboken, NJ, 2014.Google Scholar
Fathi, A.. Expansiveness, hyperbolicity and Hausdorff dimension. Comm. Math. Phys. 126 (1989), 249–262.CrossRefGoogle Scholar
Fraser, J. M.. Assouad type dimensions and homogeneity of fractals. Trans. Amer. Math. Soc. 366 (2014), 6687–6733.CrossRefGoogle Scholar
Fraser, J. M.. Regularity of Kleinian limit sets and Patterson-Sullivan measures. Trans. Amer. Math. Soc. 372 (2019), 4977–5009.CrossRefGoogle Scholar
Fraser, J. M.. Assouad Dimension and Fractal Geometry. Cambridge University Press, Cambridge, 2020.CrossRefGoogle Scholar
Fried, D.. Finitely presented dynamical systems. Ergod. Th. & Dynam. Sys. 7 (1987), 489–507.CrossRefGoogle Scholar
Goffeng, M. and Mesland, B.. Spectral triples on ${\mathbf{\mathcal{O}}}_N$ . Conf. Proc. MATRIX-Program “Refining ${C}^{\ast }$ -Algebraic Invariants for Dynamics using KK-Theory” Creswick, Australia, 2016.Google Scholar
Hamenstädt, U.. A new description of the Bowen-Margulis measure. Ergod. Th. & Dynam. Sys. 9 (1989), 455–464.CrossRefGoogle Scholar
Hasselblatt, B.. A new construction of the Margulis measure for Anosov flows. Ergod. Th. & Dynam. Sys. 9 (1989), 465–468.CrossRefGoogle Scholar
Heinonen, J.. Lectures on Analysis on Metric Spaces. Springer, New York, NY, 2001.CrossRefGoogle Scholar
Herman, R. H., Putnam, I. F. and Skau, C. F.. Ordered Bratteli diagrams, dimension groups and topological dynamics. Int. J. Math. 3 (1992), 827–864.CrossRefGoogle Scholar
Julien, A., Kellendonk, J. and Savinien, J.. On the noncommutative geometry of tilings. Mathematics of Aperiodic Order. Birkhäuser/Springer, Basel, CH, 2015, pp. 259–306.CrossRefGoogle Scholar
Käenmäki, A. and Rossi, E.. Weak separation condition, Assouad dimension, and Furstenberg homogeneity. Ann. Acad. Sci. Fenn. Math. 41 (2016), 465–490.CrossRefGoogle Scholar
Käenmäki, A. and Vilppolainen, M.. Separation conditions on controlled Moran constructions. Fund. Math. 200 (2008), 69–100.CrossRefGoogle Scholar
Kaimanovich, V. A.. Bowen-Margulis and Patterson measures on negatively curved compact manifolds. Adv. Ser. Dyn. Sys. 9 (1991), 223–232.Google Scholar
Kaminker, J., Putnam, I. F., and Whittaker, M. F.. K-theoretic duality for hyperbolic dynamical systems. J. Reine Angew. Math. 730 (2017), 263–299.Google Scholar
Katok, A. and Hasselblatt, B.. Introduction to the Modern Theory of Dynamical Systems. Cambridge University Press, Cambridge, UK, 1995.CrossRefGoogle Scholar
Lang, U. and Schlichenmaier, T.. Nagata dimension, quasisymmetric embeddings, and Lipschitz extensions. Int. Math. Res. Not. 58 (2005), 3625–3655.CrossRefGoogle Scholar
Luukkainen, J.. Assouad dimension: antifractal metrization, porous sets, and homogeneous measures. J. Korean Math. Soc. 35 (1998), 23–76.Google Scholar
Mackay, J. M.. Assouad dimension of self-affine carpets. Conform. Geom. Dyn. 15 (2011), 177–187.CrossRefGoogle Scholar
Mackay, J. M. and Tyson, J. T.. Conformal Dimension. Theory and Application. American Mathematical Society, Providence, RI, 2010.Google Scholar
Mané, R.. Expansive homeomorphisms and topological dimension. Trans. Amer. Math. Soc. 252 (1979), 313–319.CrossRefGoogle Scholar
Mané, R.. The Hausdorff dimension of horseshoes of diffeomorphisms of surfaces. Bol. Soc. Brasil. Mat. 20 (1990), 1–24.CrossRefGoogle Scholar
McCluskey, H. and Manning, A.. Hausdorff dimension for horseshoes. Ergod. Th. & Dynam. Sys. 3 (1983), 251–260.CrossRefGoogle Scholar
Nagata, J.. Modern Dimension Theory. Heldermann Verlag, Berlin, DE, 1983.Google Scholar
Naor, A. and Neiman, O.. Assouad’s theorem with dimension independent of the snowflaking. Rev. Mat. Iberoam. 28 (2012), 1123–1142.CrossRefGoogle Scholar
Nekrashevych, V.. Hyperbolic groupoids: metric and measure. Groups Geom. Dyn. 8 (2014), 883–932.CrossRefGoogle Scholar
Olsen, L.. On the Assouad dimension of graph directed Moran fractals. Fractals 19 (2011), 221–226.CrossRefGoogle Scholar
Palis, J. and Viana, M.. On the Continuity of Hausdorff Dimension and Limit Capacity for Horseshoes (Lecture Notes in Mathematics, 1331). Springer-Verlag, New York, NY, 1988, pp. 150–160.Google Scholar
Palmer, I.. Riemannian geometry of compact metric spaces. PhD Thesis, Georgia Institute of Technology, 2010.Google Scholar
Parry, W.. Intrinsic Markov chains. Trans. Amer. Math. Soc. 112 (1964), 55–66.CrossRefGoogle Scholar
Patterson, S. J.. The limit set of a Fuchsian group. Acta Math. 136 (1976), 241–273.CrossRefGoogle Scholar
Pearson, J. and Bellissard, J.. Noncommutative Riemannian geometry and diffusion on ultrametric Cantor sets. J. Noncommut. Geom. 3 (2009), 447–480.CrossRefGoogle Scholar
Pesin, Y.. Dimension Theory in Dynamical Systems: Contemporary Views and Applications. Chicago University Press, Chicago, IL, 1997.CrossRefGoogle Scholar
Pesin, Y. and Weiss, H.. Multifractal analysis of equilibrium measures for conformal expanding maps and Moran-like geometric constructions. J. Stat. Phys. 86 (1997), 233–275.CrossRefGoogle Scholar
Pollicott, M. and Weiss, H.. The dimensions of some self-affine limit sets in the plane and hyperbolic sets. J. Stat. Phys. 77 (1994), 841–866.CrossRefGoogle Scholar
Proietti, V.. A note on homology for Smale spaces. Groups Geom. Dyn. 14 (2020), 813–836.CrossRefGoogle Scholar
Putnam, I. F.. ${C}^{\ast }$ -algebras from Smale spaces. Canad. J. Math. 48 (1996), 175–195.CrossRefGoogle Scholar
Putnam, I. F.. A Homology Theory for Smale Spaces. Vol. 232. American Mathematical Society, Providence, RI, 2014, pp. viii+122.Google Scholar
Putnam, I. F.. Smale spaces and ${C}^{\ast }$ -algebras. Lecture Notes, University of Victoria, 2006.Google Scholar
Putnam, I. F. and Spielberg, J.. The structure of ${C}^{\ast }$ -algebras associated with hyperbolic dynamical systems. J. Funct. Anal. 163 (1999), 279–299.CrossRefGoogle Scholar
Robinson, J.. Dimensions, Embeddings, and Attractors. Cambridge University Press, Cambridge, UK, 2010.CrossRefGoogle Scholar
Ruelle, D.. Repellers for real analytic maps. Ergod. Th. & Dynam. Sys. . 2 (1982), 99–108.CrossRefGoogle Scholar
Ruelle, D.. Thermodynamic Formalism. Cambridge University Press, Cambridge, UK, 2004.CrossRefGoogle Scholar
Ruelle, D. and Sullivan, D.. Currents, flows, and diffeomorphisms. Topology 14 (1975), 319–327.CrossRefGoogle Scholar
Sakai, K.. Shadowing properties of $L$ -hyperbolic homeomorphisms. Topology Appl. 112 (2001), 229–243.CrossRefGoogle Scholar
Samko, S. G., Kilbas, A. A. and Marichev, O. I.. Fractional Integrals and Derivatives: Theory and Applications. Gordon and Breach Science Publications, New York, NY and London, UK, 1993.Google Scholar
Schoenfeld, A. H.. Continuous surjections from cantor sets to compact metric spaces. Proc. Amer. Math. Soc. 46 (1974), 141–142.CrossRefGoogle Scholar
Semmes, S.. On the nonexistence of bilipschitz parametrizations and geometric problems about ${A}^{\infty }$ -weights. Rev. Mat. Iberoam. 12 (1996), 337–410.CrossRefGoogle Scholar
Simon, K. and Solomyak, B.. Hausdorff dimension for horseshoes in ${\mathbb{R}}^3$ . Ergod. Th. & Dynam. Sys. 19 (1999), 1343–1363.CrossRefGoogle Scholar
Smale, S.. Differentiable dynamical systems. Bull. Amer. Math. Soc. 73 (1967), 747–817.CrossRefGoogle Scholar
Sullivan, D.. Entropy, Hausdorff measures old and new, and limit sets of geometrically finite Kleinian groups. Acta Math. 153 (1984), 259–277.CrossRefGoogle Scholar
Takens, F.. Limit Capacity on Dynamically Defined Cantor Sets (Lecture Notes in Mathematics, 1331). Springer-Verlag, New York, NY, 1988, pp. 196–212.Google Scholar
Walters, P.. An Introduction to Ergodic Theory. Springer, New York, NY, 1982.CrossRefGoogle Scholar
Wieler, S.. Smale spaces with totally disconnected local stable sets. PhD Thesis, University of Victoria, 2012.Google Scholar
Williams, R. F.. Expanding attractors. Publ. Math. Inst. Hautes Études Sci. 43 (1974), 169–203.CrossRefGoogle Scholar
Zähle, M.. Harmonic calculus on fractals, a measure geometric approach. II. Trans. Amer. Math. Soc. 357 (2005), 3407–3423.CrossRefGoogle Scholar