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Diophantine approximation by orbits of expanding Markov maps

Published online by Cambridge University Press:  07 February 2012

LINGMIN LIAO  and
STÉPHANE SEURET
LINGMIN LIAO
Affiliation:
LAMA, CNRS UMR 8050, Université Paris-Est Créteil, 61 Avenue du Général de Gaulle, 94010 Créteil Cedex, France (email: lingmin.liao@u-pec.fr, seuret@u-pec.fr)
STÉPHANE SEURET
Affiliation:
LAMA, CNRS UMR 8050, Université Paris-Est Créteil, 61 Avenue du Général de Gaulle, 94010 Créteil Cedex, France (email: lingmin.liao@u-pec.fr, seuret@u-pec.fr)
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Abstract

In 1995, Hill and Velani introduced the ‘shrinking targets’ theory. Given a dynamical system ([0,1],T), they investigated the Hausdorff dimension of sets of points whose orbits are close to some fixed point. In this paper, we study the sets of points well approximated by orbits {Tnx}n≥0, where Tis an expanding Markov map with a finite partition supported by [0,1]. The dimensions of these sets are described using the multifractal properties of invariant Gibbs measures.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 33 , Issue 2 , April 2013 , pp. 585 - 608
Copyright
©2012 Cambridge University Press

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References

[1]Baladi, V.. Positive Transfer Operators and Decay of Correlations (Advanced Series in Nonlinear Dynamics, 16). World Scientific, River Edge, NJ, 2000.Google Scholar
[2]Barral, J., Ben Nasr, F. and Peyrière, J.. Comparing multifractal formalisms: the neighboring boxes conditions. Asian J. Math. 7 (2003), 149–165.Google Scholar
[3]Barral, J. and Seuret, S.. Heterogeneous ubiquitous systems in ℝd and Hausdorff dimension. Bull. Braz. Math. Soc. (N.S.) 38(3) (2007), 467–515.Google Scholar
[4]Barral, J. and Seuret, S.. Ubiquity and large intersections properties under digit frequencies constraints. Math. Proc. Cambridge Philos. Soc. 145(3) (2008), 527–548.CrossRefGoogle Scholar
[5]Barreira, L., Pesin, Y. and Schmeling, J.. On a general concept of multifractality: multifractal spectra for dimensions, entropies, and Lyapunov exponents. Multifractal rigidity. Chaos 7 (1997), 27–38.Google Scholar
[6]Beresnevich, V. and Velani, S.. A Mass Transference Principle and the Duffin–Schaeffer conjecture for Hausdorff measures. Ann. of Math. (2) 164(3) (2006), 971–992.Google Scholar
[7]Besicovitch, A. S.. Sets of fractional dimension (IV): on rational approximation to real numbers. J. Lond. Math. Soc. 9 (1934), 126–131.Google Scholar
[8]Bowen, R.. Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms. Springer, Berlin, 1975.Google Scholar
[9]Brown, G., Michon, G. and Peyrière, J.. On the multifractal analysis of measures. J. Stat. Phys. 66 (1992), 775–790.CrossRefGoogle Scholar
[10]Bugeaud, Y.. Approximation by algebraic integers and Hausdorff dimension. J. Lond. Math. Soc. (2) 65 (2002), 547–559.Google Scholar
[11]Bugeaud, Y.. A note on inhomogeneous diophantine approximation. Glasg. Math. J. 45 (2003), 105–110.CrossRefGoogle Scholar
[12]Bugeaud, Y., Harrap, S., Kristensen, S. and Velani, S.. On shrinking targets for ℤm-actions on the torii. Mathematika 56 (2010), 193–202.CrossRefGoogle Scholar
[13]Cassels, J. W. S.. An introduction to diophantine approximation (Cambridge Tracts in Mathematics and Mathematical Physics, 45). Cambridge University Press, New York, 1957.Google Scholar
[14]Collet, P., Lebowitz, J. and Porzio, A.. The dimension spectrum of some dynamical systems. J. Stat. Phys. 47 (1987), 609–644.Google Scholar
[15]Dodson, M. M., Melián, M. V., Pestana, D. and Velani, S. L.. Patterson measure and Ubiquity. Ann. Acad. Sci. Fenn. Ser. A I Math. 20 (1995), 37–60.Google Scholar
[16]Einsiedler, M., Katok, A. and Lindenstrauss, E.. Invariant measures and the set of exceptions to Littlewood’s conjecture. Ann. of Math. (2) 164(2) (2006), 513–560.Google Scholar
[17]Falconer, K. J.. Fractal geometry. Mathematical Foundations and Applications, 2nd edn. John Wiley, Hoboken, NJ, 2003.Google Scholar
[18]Fan, A.-H., Schmeling, J. and Troubetzkoy, S.. Dynamical Diophantine approximation. Preprint, 2009.Google Scholar
[19]Galatolo, S.. Dimension and hitting time in rapidly mixing systems. Math. Res. Lett. 14(5) (2007), 797–805.Google Scholar
[20]Hill, R. and Velani, S. L.. Ergodic theory of shrinking targets. Invent. Math. 119 (1995), 175–198.Google Scholar
[21]Hill, R. and Velani, S. L.. The shrinking target problem for matrix transformations of tori. J. Lond. Math. Soc. (2) 60(2) (1999), 381–398.Google Scholar
[22]Jarnik, V.. Diophantische Approximationen und Hausdorffsches Mass. Mat. Sb. 36 (1929), 371–381.Google Scholar
[23]Kim, D. H.. The shrinking target property of irrational rotations. Nonlinearity 20(7) (2007), 1637–1643.Google Scholar
[24]Kleinbock, D. and Margulis, G. A.. Flows on homogeneous spaces and Diophantine approximation on manifolds. Ann. of Math. (2) 148 (1998), 339–360.Google Scholar
[25]Kleinbock, D., Lindenstrauss, E. and Weiss, B.. On fractal measures and Diophantine approximation. Selecta Math. (N.S.) 10 (2004), 479–523.Google Scholar
[26]Liverani, C., Saussol, B. and Vaienti, S.. Conformal measure and decay of correlation for covering weighted systems. Ergod. Th. & Dynam. Sys. 18(6) (1998), 1399–1420.CrossRefGoogle Scholar
[27]Ornstein, D. and Weiss, B.. Entropy and data compression schemes. IEEE Trans. Inform. Theory 39(1) (1993), 78–83.Google Scholar
[28]Pesin, Y. and Weiss, H.. The multifractal analysis of Gibbs measures: motivation, mathematical foundation, and examples. Chaos 7(1) (1997), 89–106.Google Scholar
[29]Parry, W. and Pollicott, M.. Zeta functions and the periodic orbit structure of hyperbolic dynamics. Astérisque 187–188 (1990).Google Scholar
[30]Philipp, W. and Stout, W.. Almost sure invariance principles for partial sums of weakly dependent random variables. Mem. Amer. Math. Soc. 2 161 (1975).Google Scholar
[31]Rand, D. A.. The singularity spectrum f(α) for cookie-cutters. Ergod. Th. & Dynam. Sys. 9(3) (1989), 527–541.Google Scholar
[32]Ruelle, D.. Thermodynamic formalism. The Mathematical Structures of Equilibrium Statistical Mechanics, 2nd edn(Cambridge Mathematical Library). Cambridge University Press, Cambridge, 2004.Google Scholar
[33]Schmeling, J. and Troubetzkoy, S.. Inhomogeneous Diophantine approximation and angular recurrence properties of the billiard flow in certain polygons. Mat. Sb. 194 (2003), 295–309.Google Scholar
[34]Simpelaere, D.. Dimension spectrum of Axiom A diffeomorphisms. II. Gibbs measures. J. Stat. Phys. 76(5–6) (1994), 1359–1375.Google Scholar
[35]Walters, P.. Invariant measures and equilibrium states for some mappings which expand distances. Trans. Amer. Math. Soc. 236 (1978), 121–153.CrossRefGoogle Scholar