STABILITY AND PERTURBATIONS OF COUNTABLE MARKOV MAPS

Let T and , , be countable Markov maps such that the branches of converge pointwise to the branches of T, as . We study the stability of various quantities measuring the singularity (dimension, Holder exponent etc) of the topological conjugacy between and T when . This is a well-understood problem for maps with finitely-many branches, and the quantities are stable for small e, that is, they converge to their expected values if . For the infinite branch case their stability might be expected to fail, but we prove that even in the infinite branch case the quantity is stable under some natural regularity assumptions on and T (under which, for instance, the Holder exponent of fails to be stable). Our assumptions apply for example in the case of Gauss map, various Luroth maps and accelerated Manneville–Pomeau maps when varying the parameter α. For the proof we introduce a mass transportation method from the cusp that allows us to exploit thermodynamical ideas from the finite branch case.