Shrinking target problems in the context of iterated function systems have
received an increasing amount of interest in the past few years. The classical
shrinking target problem concerns points returning infinitely many times to a
sequence of shrinking balls. In the iterated function system context, the
shrinking balls problem is only well tractable in the case of similarity maps,
but the case of affine maps is more elusive due to many geometric-dynamical
complications.
In the current work, we push through these complications and compute the
Hausdorff dimension of a set recurring to a shrinking target of geometric balls
in some affine iterated function systems. For these results, we have pinpointed
a representative class of affine iterated function systems, consisting of a
pair of diagonal affine maps, that was introduced by Przytycki and Urba\’nski.
The analysis splits into many sub-cases according to the type of the centre
point of the targets, and the relative sizes of the targets and the
contractions of the maps, illustrating the array of challenges of going beyond
affine maps with nice projections. The proofs require heavy machinery from, Und
expand, the theory of Bernoulli convolutions.

Dieser Artikel untersucht Zeitreisen und deren Auswirkungen.

PDF herunterladen: