Stability and perturbations of countable Markov maps

Let T and Tϵ, ϵ > 0, be countable Markov maps such that the branches of Tϵ converge pointwise to the branches of T, as ϵ → 0. We study the stability of various quantities measuring the singularity (dimension, Hölder exponent etc) of the topological conjugacy θ ϵ between Tϵ and T when ϵ → 0. This is a wellunderstood problem for maps with finitely-many branches, and the quantities are stable for small ϵ, that is, they converge to their expected values if ϵ → 0. For the infinite branch case their stability might be expected to fail, but we prove that even in the infinite branch case the quantity dimH{x : θ′ ϵ (x) ≠ 0} is stable under some natural regularity assumptions on Tϵ and T (under which, for instance, the Hölder exponent of θϵ fails to be stable). Our assumptions apply for example in the case of Gauss map, various Löroth maps and accelerated Manneville-Pomeau maps x → x + x1+α mod 1 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.