## 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.

Original language | English |
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Pages (from-to) | 1351-1377 |

Number of pages | 27 |

Journal | Nonlinearity |

Volume | 31 |

Issue number | 4 |

Early online date | 27 Feb 2018 |

DOIs | |

Publication status | Published - 27 Feb 2018 |

## Keywords

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