Metadata
Abstract
Generalising a construction of Falconer, we consider classes of -subsets of with the property that sets belonging to the class have large Hausdorff dimension and the class is closed under countable intersections. We relate these classes to some inhomogeneous potentials and energies, thereby providing some useful tools to determine if a set belongs to one of the classes.
As applications of this theory, we calculate, or at least estimate, the Hausdorff dimension of randomly generated limsup-sets, and sets that appear in the setting of shrinking targets in dynamical systems. For instance, we prove that for ,
for almost every , where is a quadratic map with in a set of parameters described by Benedicks and Carleson.
References
[AP19] Shrinking targets in parametrised families, Math. Proc. Camb. Philos. Soc., Volume 166 (2019) no. 2, pp. 265-295 | DOI | MR | Zbl
[BC91] The dynamics of the Hénon map, Ann. Math., Volume 133 (1991) no. 1, pp. 73-169 | DOI | Zbl
[BL16] Uniform diophantine approximation related to -ary and -expansions, Ergodic Theory Dyn. Syst., Volume 36 (2016) no. 1, pp. 1-22 | DOI | MR | Zbl
[BW14] Distribution of full cylinders and the diophantine properties of the orbits in -expansions, J. Fractal Geom., Volume 1 (2014) no. 2, pp. 221-241 | DOI | MR | Zbl
[Dur10] On randomly placed arcs on the circle, Recent developments in fractals and related fields (Applied and Numerical Harmonic Analysis), Birkhäuser, 2010, 343–351 pages | DOI | MR | Zbl
[EP18] Hausdorff dimension of random limsup sets, J. Lond. Math. Soc., Volume 98 (2018) no. 3, pp. 661-686 | DOI | MR | Zbl
[Fal85] Classes of sets with large intersection, Mathematika, Volume 32 (1985), pp. 191-205 | DOI | MR | Zbl
[Fal94] Sets with large intersection properties, J. Lond. Math. Soc., Volume 49 (1994) no. 2, pp. 267-280 | DOI | MR | Zbl
[Fal97] Techniques in fractal geometry, John Wiley & Sons, 1997, xvii+256 pages | Zbl
[FFW01] Recurrence, dimension and entropy, J. Lond. Math. Soc., Volume 64 (2001) no. 1, pp. 229-244 | DOI | MR | Zbl
[FJJS18] Dimensions of random covering sets in Riemann manifolds, Ann. Probab., Volume 46 (2018) no. 3, pp. 1201-1805 | MR | Zbl
[Fro35] Potentiel d’équilibre et capacité des ensembles avec quelques applications a la théorie des fonctions, Meddelanden från Lunds Universitets Matematiska Seminarium, 3, 1935 | Zbl
[FST13] A multifractal mass transference principle for Gibbs measures with applications to dynamical diophantine approximation, Proc. Lond. Math. Soc., Volume 107 (2013) no. 5, pp. 1173-1219 | DOI | MR | Zbl
[FW04] On the covering by small random intervals, Ann. Inst. Henri Poincaré, Probab. Stat., Volume 40 (2004) no. 1, pp. 125-131 | DOI | Numdam | MR | Zbl
[HV95] The ergodic theory of shrinking targets, Invent. Math., Volume 119 (1995) no. 1, pp. 175-198 | DOI | MR | Zbl
[JJK + 14] Hausdorff dimension of affine random covering sets in torus, Ann. Inst. Henri Poincaré, Probab. Stat., Volume 50 (2014) no. 4, pp. 1371-1384 | DOI | Numdam | MR | Zbl
[LS13] Diophantine approximation by orbits of expanding Markov maps, Ergodic Theory Dyn. Syst., Volume 33 (2013) no. 2, pp. 585-608 | DOI | MR | Zbl
[LSV98] Conformal measure and decay of correlation for covering weighted systems, Ergodic Theory Dyn. Syst., Volume 18 (1998) no. 6, pp. 1399-1420 | DOI | MR | Zbl
[LWWX14] The shrinking target problem in the dynamical system of continued fractions, Proc. Lond. Math. Soc., Volume 108 (2014) no. 1, pp. 159-186 | MR | Zbl
[Ols95] A multifractal formalism, Adv. Math., Volume 116 (1995) no. 1, pp. 82-196 | DOI | MR | Zbl
[Per15] A note on random coverings of tori, Bull. Lond. Math. Soc., Volume 47 (2015) no. 1, pp. 7-12 | DOI | MR | Zbl
[PR15] A Frostman-type lemma for sets with large intersections, and an application to diophantine approximation, Proc. Edinb. Math. Soc., Volume 58 (2015) no. 2, pp. 521-542 | DOI | MR | Zbl
[PR17] On shrinking targets for piecewise expanding interval maps, Ergodic Theory Dyn. Syst., Volume 37 (2017) no. 2, pp. 646-663 | DOI | MR | Zbl
[Rog70] Hausdorff measures, Cambridge University Press, 1970 | Zbl
[Seu18] Inhomogeneous coverings of topological Markov shifts, Math. Proc. Camb. Philos. Soc., Volume 165 (2018) no. 2, pp. 341-357 | DOI | MR | Zbl
[You92] Decay of correlations for certain quadratic maps, Commun. Math. Phys., Volume 146 (1992) no. 1, pp. 123-138 | DOI | MR | Zbl