Stability and perturbations of countable Markov maps
| Autor: | Tuomas Sahlsten, Sara Munday, Thomas Jordan |
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| Jazyk: | angličtina |
| Rok vydání: | 2018 |
| Předmět: |
Pure mathematics
Gauss map Differentiability General Physics and Astronomy Hausdorff dimension Physics and Astronomy(all) Perturbations 01 natural sciences Stability (probability) Singularity Countable Markov maps Countable set 0101 mathematics Mathematical Physics Mathematics Pointwise Markov chain Applied Mathematics 010102 general mathematics Statistical and Nonlinear Physics 010101 applied mathematics Topological conjugacy Non-uniformly hyperbolic dynamics Thermodynamical formalism |
| Zdroj: | Jordan, T, Munday, S & Sahlsten, T 2018, ' Stability and perturbations of countable Markov maps ', Nonlinearity, vol. 31, no. 4, pp. 1351-1377 . https://doi.org/10.1088/1361-6544/aa9d5b |
| DOI: | 10.1088/1361-6544/aa9d5b |
| Popis: | 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. |
| Databáze: | OpenAIRE |
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