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Publication - Dr Thomas Jordan

    Stability and perturbations of countable Markov maps


    Jordan, T, Munday, S & Sahlsten, T, 2017, ‘Stability and perturbations of countable Markov maps’. Nonlinearity.


    Abstract. Let T and T", " > 0, be countable Markov maps such that the branches of T" converge
    pointwise to the branches of T, as " ! 0. We study the stability of various quantities measuring the
    singularity (dimension, H¨older exponent etc.) of the topological conjugacy ✓" between T" and T when " ! 0. This is a well-understood problem for maps with finitely-many branches, and the quantities are stable for small ", that is, they converge to their expected values if " ! 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 : ✓0 "(x) 6= 0} is stable under some natural regularity assumptions on T" and T (under which, for instance, the H¨older exponent of ✓" fails to be stable). Our assumptions apply for example in the case of Gauss map, various L¨uroth maps and accelerated Manneville-Pomeau maps x 7! x + x1+↵ mod 1 when varying the parameter ↵. For the proof we introduce a mass transportation method from the cusp that allows us to exploit thermodynamical ideas from the finite branch case.

    Full details in the University publications repository