Inhomogeneous potentials, Hausdorff dimension and shrinking targets

Persson, Tomas

doi:10.5802/ahl.15

Persson, Tomas

Inhomogeneous potentials, Hausdorff dimension and shrinking targets

Annales Henri Lebesgue, Volume 2 (2019), pp. 1-37.

Metadata

DOI10.5802/ahl.15

Keywords Hausdorff dimension, limsup sets, potentials

Abstract

Generalising a construction of Falconer, we consider classes of $G_{δ}$ -subsets of $ℝ^{d}$ 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 $α \geq 1$ ,

\dim_{H} {y : | T_{a}^{n} (x) - y | < n^{- α} infinitely often} = \frac{1}{α},

for almost every $x \in [1 - a, 1]$ , where $T_{a}$ is a quadratic map with $a$ in a set of parameters described by Benedicks and Carleson.

References

[AP19] Aspenberg, Magnus; Persson, Tomas Shrinking targets in parametrised families, Math. Proc. Camb. Philos. Soc., Volume 166 (2019) no. 2, pp. 265-295 | DOI | MR | Zbl

[BC91] Benedicks, Michael; Carleson, Lennart The dynamics of the Hénon map, Ann. Math., Volume 133 (1991) no. 1, pp. 73-169 | DOI | Zbl

[BL16] Bugeaud, Yann; Liao, Lingmin Uniform diophantine approximation related to $b$ -ary and $β$ -expansions, Ergodic Theory Dyn. Syst., Volume 36 (2016) no. 1, pp. 1-22 | DOI | MR | Zbl

[BW14] Bugeaud, Yann; Wang, Bao-Wei Distribution of full cylinders and the diophantine properties of the orbits in $β$ -expansions, J. Fractal Geom., Volume 1 (2014) no. 2, pp. 221-241 | DOI | MR | Zbl

[Dur10] Durand, Arnaud 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] Ekström, Fredrik; Persson, Tomas Hausdorff dimension of random limsup sets, J. Lond. Math. Soc., Volume 98 (2018) no. 3, pp. 661-686 | DOI | MR | Zbl

[Fal85] Falconer, Kenneth J. Classes of sets with large intersection, Mathematika, Volume 32 (1985), pp. 191-205 | DOI | MR | Zbl

[Fal94] Falconer, Kenneth J. Sets with large intersection properties, J. Lond. Math. Soc., Volume 49 (1994) no. 2, pp. 267-280 | DOI | MR | Zbl

[Fal97] Falconer, Kenneth J. Techniques in fractal geometry, John Wiley & Sons, 1997, xvii+256 pages | Zbl

[FFW01] Fan, Ai-Hua; Feng, De-Jun; Wu, Jun Recurrence, dimension and entropy, J. Lond. Math. Soc., Volume 64 (2001) no. 1, pp. 229-244 | DOI | MR | Zbl

[FJJS18] Feng, De-Jun; Järvenpää, Esa; Järvenpää, Maarit; Suomala, Ville Dimensions of random covering sets in Riemann manifolds, Ann. Probab., Volume 46 (2018) no. 3, pp. 1201-1805 | MR | Zbl

[Fro35] Frostman, Otto 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] Fan, Ai-Hua; Schmeling, Jörg; Troubetzkoy, Serge 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] Fan, Ai-Hua; Wu, Jun 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] Hill, Richard; Velani, Sanju L. The ergodic theory of shrinking targets, Invent. Math., Volume 119 (1995) no. 1, pp. 175-198 | DOI | MR | Zbl

[JJK + 14] Järvenpää, Esa; Järvenpää, Maarit; Koivusalo, Henna; Li, Bing; Suomala, Ville 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] Liao, Lingmin; Seuret, Stéphane Diophantine approximation by orbits of expanding Markov maps, Ergodic Theory Dyn. Syst., Volume 33 (2013) no. 2, pp. 585-608 | DOI | MR | Zbl

[LSV98] Liverani, Carlangelo; Soussol, Benoit; Vaienti, Sandro 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] Li, Bing; Wang, Bao-Wei; Wu, Jun; Xu, Jian 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] Olsen, Lars A multifractal formalism, Adv. Math., Volume 116 (1995) no. 1, pp. 82-196 | DOI | MR | Zbl

[Per15] Persson, Tomas A note on random coverings of tori, Bull. Lond. Math. Soc., Volume 47 (2015) no. 1, pp. 7-12 | DOI | MR | Zbl

[PR15] Persson, Tomas; Reeve, Henry W. J. 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] Persson, Tomas; Rams, Michał On shrinking targets for piecewise expanding interval maps, Ergodic Theory Dyn. Syst., Volume 37 (2017) no. 2, pp. 646-663 | DOI | MR | Zbl

[Rog70] Rogers, Claude A. Hausdorff measures, Cambridge University Press, 1970 | Zbl

[Seu18] Seuret, Stéphane Inhomogeneous coverings of topological Markov shifts, Math. Proc. Camb. Philos. Soc., Volume 165 (2018) no. 2, pp. 341-357 | DOI | MR | Zbl

[You92] Young, Lai-Sang Decay of correlations for certain quadratic maps, Commun. Math. Phys., Volume 146 (1992) no. 1, pp. 123-138 | DOI | MR | Zbl

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