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Stability and perturbations of countable Markov maps

Research output: Contribution to journalArticle

Original languageEnglish
Pages (from-to)1351-1377
Number of pages27
JournalNonlinearity
Volume31
Issue number4
DOIs
DateAccepted/In press - 27 Nov 2017
DatePublished (current) - 27 Feb 2018

Abstract

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.

    Research areas

  • Countable Markov maps, Differentiability, Hausdorff dimension, Non-uniformly hyperbolic dynamics, Perturbations, Thermodynamical formalism

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  • Full-text PDF (accepted author manuscript)

    Rights statement: This is the author accepted manuscript (AAM). The final published version (version of record) is available online via IOP at http://iopscience.iop.org/article/10.1088/1361-6544/aa9d5b/meta. Please refer to any applicable terms of use of the publisher.

    Accepted author manuscript, 789 KB, PDF document

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