Expanding maps, shrinking targets and hitting times Articles uri icon

publication date

  • September 2012

start page

  • 2443

end page

  • 2471

volume

  • 25

International Standard Serial Number (ISSN)

  • 0951-7715

Electronic International Standard Serial Number (EISSN)

  • 1361-6544

abstract

  • We study the (metric) Diophantine approximation properties of uniformly expanding transformations and some non-uniformly expanding transformations, i.e. transformations T (x) with an associated countable (not necessarily finite) partition and a return time function R(x) (constant on the blocks of the partition) so that (T) over cap (x) = T-R(x) (x) is uniformly expanding, and we obtain Borel-Cantelli results on hitting times of shrinking targets. Our arguments do not require the so-called big image property for (T) over cap and our results contain most of the diversity of examples of slowly mixing systems. We also obtain, with related techniques, results for one-sided topological Markov chains over a countable alphabet with a Gibbs measure.