Title: Hausdorff and packing measure for limit sets of conformal repellers and iterated function systems.

Abstract:

In this talk, I will present some recent results concerning the (numerical value of) Hausdorff and packing measures (Hh and Ph) on limit sets of a class of conformal iterated function systems or conformal repellers

For such systems, the way of finding exact value of Hausdorff/packing dimension was established long time ago (versions of Bowen's formula). Also, a lot is known about dependence of the dimension on parameter in naturally parametrized families of such systems.

Studying the numerical value of Hausdorff/packing measure of the limit set (evaluated in its Hausdorff/packing dimension) is a challenging problem. There is no visible way of determining the value of Hausdorff (or: packing) measure in terms of thermodynamic formalism.

In our joint paper with Mariusz Urbański, we considered the sequence of iterated function systems approximating the Gauss map. The limit set Jn of such n-th systems is exactly the set of irrational numbers in [0,1] whose continued fraction expansion has entries bounded by n.

Denote by hn the Hausdorff dimension of Jn. The asymptotic of hn is known (Hensley), namely:

(1-hn)⋅n→6π2 as n→∞.

In the above mentioned work with Mariusz, we proved continuity of Hausdorff measure: Hhn(Jn)→1 as n→∞.

In the present joint work with Mariusz Urbański and Rafał Tryniecki, we study much more delicate question, namely -asymptotic of the value 1-Hhn(Jn). We do it for the Gauss map and its piecewise linear analogue.

I will also mention some recent results of R Tryniecki concerning analogous questions of continuity of Hausdorff and packing measure for other classes of infinite iterated function systems.