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.
[BC91] The dynamics of the Hénon map, Ann. Math., Volume 133 (1991) no. 1, pp. 73-169 | Zbl 0724.58042
[Dur10] On randomly placed arcs on the circle, Recent developments in fractals and related fields, Birkhäuser (Applied and Numerical Harmonic Analysis) (2010), 343–351 pages | Article | Zbl 1195.28001
[Fal97] Techniques in fractal geometry, John Wiley & Sons (1997), xvii+256 pages | Zbl 0869.28003
[Fro35] Potentiel d’équilibre et capacité des ensembles avec quelques applications a la théorie des fonctions, Meddelanden från Lunds Universitets Matematiska Seminarium, Volume 3 (1935) | Zbl 61.1262.02
[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 | Article | MR 3126394 | Zbl 1347.37023
[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 | Article | MR 3341452 | Zbl 1323.11054
[Rog70] Hausdorff measures, Cambridge University Press (1970) | Zbl 0204.37601