On the intersection of dynamical covering sets with fractals

Let (X,B, μ, T, d) be a measure-preserving dynamical system with exponentially mixing property, and let μ be an Ahlfors s-regular probability measure. The dynamical covering problem concerns the set E(x) of points which are covered by the orbits of x ∈ X infinitely many times. We prove that the Hausdorff dimension of the intersection of E(x) and any regular fractal G with dimH G > s−α equals dimH G+α−s, where α = dimH E(x) μ–a.e. Moreover, we obtain the packing dimension of E(x)∩G and an estimate for dimH(E(x) ∩G) for any analytic set G.

[1]  Bing Li,et al.  Hitting probabilities of random covering sets in tori and metric spaces , 2015, 1510.06630.

[2]  Han Yu,et al.  Rational points near self-similar sets , 2021, 2101.05910.

[3]  Jian Xu,et al.  Dynamical covering problems on the triadic Cantor set , 2017 .

[4]  Arnaud Durand.  On randomly placed arcs on the circle , 2010 .

[5]  Y. Peres,et al.  Limsup Random Fractals , 2000 .

[6]  T. Persson,et al.  On shrinking targets for piecewise expanding interval maps , 2014, Ergodic Theory and Dynamical Systems.

[7]  Jian Xu,et al.  The shrinking target problem in the dynamical system of continued fractions , 2014 .

[8]  A note on the hitting probabilities of random covering sets , 2013, 1307.2819.

[9]  C. Tricot.  Two definitions of fractional dimension , 1982, Mathematical Proceedings of the Cambridge Philosophical Society.

[10]  T. Eisner,et al.  Ergodic Theorems , 2019, Probability.

[11]  On a problem of K. Mahler: Diophantine approximation and Cantor sets , 2005, math/0505074.

[12]  Jun Wu,et al.  On the covering by small random intervals , 2004 .

[13]  Walter Philipp,et al.  Some metrical theorems in number theory , 1967 .

[14]  Ville Suomala,et al.  Existence of doubling measures via generalised nested cubes , 2010, 1011.0683.

[15]  A. Rényi.  Representations for real numbers and their ergodic properties , 1957 .

[16]  Shrinking targets in parametrised families , 2016, Mathematical Proceedings of the Cambridge Philosophical Society.

[17]  Arnaud Durand,et al.  Metric Diophantine approximation on the middle-third Cantor set , 2013, 1305.6501.

[18]  Ai-Hua Fan,et al.  A multifractal mass transference principle for Gibbs measures with applications to dynamical Diophantine approximation , 2013 .

[19]  David Simmons,et al.  Dynamical Borel–Cantelli lemma for recurrence theory , 2020, Ergodic Theory and Dynamical Systems.

[20]  K. Mahler.  Some suggestions for further research , 1984, Bulletin of the Australian Mathematical Society.

[21]  Arnaud Durand.  Sets with large intersection and ubiquity , 2008, Mathematical Proceedings of the Cambridge Philosophical Society.

[22]  Yimin Xiao,et al.  Hitting Probabilities of the Random Covering Sets , 2013 .

[23]  J. Schmeling,et al.  DISTRIBUTION OF FULL CYLINDERS AND THE DIOPHANTINE PROPERTIES OF THE ORBITS IN β-EXPANSIONS , 2014 .

[24]  J. Howroyd.  On Dimension and on the Existence of Sets of Finite Positive Hausdorff Measure , 1995 .

[25]  Harry Furstenberg,et al.  Recurrence in Ergodic Theory and Combinatorial Number Theory , 2014 .

[26]  É. Borel.  Sur les séries de Taylor , 1897 .

[27]  Ville Suomala,et al.  Dimensions of random covering sets in Riemann manifolds , 2015, 1508.07881.

[28]  R. Lyons.  Random Walks and Percolation on Trees , 1990 .

[29]  David Preiss,et al.  On the existence of subsets of finite positive packing measure , 1995 .

[30]  Box and Packing Dimensions of Typical Compact Sets , 2000 .

[31]  A Dvoretzky,et al.  ON COVERING A CIRCLE BY RANDOMLY PLACED ARCS. , 1956, Proceedings of the National Academy of Sciences of the United States of America.

[32]  C. Ryll-Nardzewski.  On the ergodic theorems (II) (Ergodic theory of continued fractions) , 1951 .

[33]  M. Boshernitzan,et al.  Quantitative recurrence results , 1993 .

[34]  Bing Li,et al.  Random covering sets in metric space with exponentially mixing property , 2020, 2009.04079.

[35]  L. Shepp.  Covering the circle with random ARCS , 1972 .

[36]  J. Kahane.  Random Coverings and Multiplicative Processes , 2000 .

[37]  Y. Peres.  Intersection-equivalence of Brownian paths and certain branching processes , 1996 .

[38]  Sanju Velani,et al.  The ergodic theory of shrinking targets , 1995 .

[39]  Stéphane Seuret,et al.  Diophantine approximation by orbits of expanding Markov maps , 2012, Ergodic Theory and Dynamical Systems.

[40]  A. Fan,et al.  Quantitative Uniform Hitting in Exponentially Mixing Systems , 2010 .

[41]  Bing Li,et al.  Hausdorff dimension of affine random covering sets in torus , 2012, Annales de l'Institut Henri Poincaré, Probabilités et Statistiques.

[42]  A. Haynes,et al.  A randomized version of the Littlewood Conjecture , 2016, 1610.08007.