Quantitative recurrence properties in conformal iterated function systems

Abstract

We consider a quantitative version of Poincaré's recurrence theorem in a conformal iterated function system. Let Φ={ϕi:i∈Λ} be a conformal iterated function system on [0,1]d with Λ a countable index set. Denote by J the attractor of Φ. Let f:[0,1]d→R+ be a positive function, Snf(x) be the sum f(x)+f(ϕw1−1(x))+⋯+f((ϕw1∘⋯∘ϕwn−1)−1(x)) (analogous to an ergodic sum), and consider the set of points for which the inequality{x∈ϕw1∘⋯∘ϕwn([0,1]d):|x−(ϕw1∘⋯∘ϕwn)−1(x)|<e−Snf(x)}, is realized for infinitely many n∈N with (w1,⋯,wn)∈Λn.The set above contains the points x whose ‘orbit’ returns very close to x, infinitely many times, with the quality of approximation depending on the time n and the point x. This can be viewed as a quantitative version of Poincaré's recurrence theorem. It is shown that its Hausdorff dimension is the solution to some pressure function. The setting considered in this paper includes the quantitative recurrence properties in the dynamical systems of b-adic expansions, continued fraction expansion, as well as some dynamical systems defined on fractal sets. Applying the main result to Diophantine approximation gives a (partial) answer to a question by K. Mahler.