Positive Lyapunov exponent for random perturbations of predominantly expanding multimodal circle maps

Blumenthal, Alex; Yang, Yun

doi:10.4171/aihpc/11

Geodesic

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Positive Lyapunov exponent for random perturbations of predominantly expanding multimodal circle maps

Blumenthal, Alex ; Yang, Yun

Annales de l'I.H.P. Analyse non linéaire, Tome 39 (2022) no. 2, pp. 419-455

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Résumé

We study the effects of independent, identically distributed random perturbations of amplitude $ε > 0$ on the asymptotic dynamics of one-parameter families ${f_{a} : S^{1} \to S^{1}, a \in [0, 1]}$ of smooth multimodal maps which are “predominantly expanding”, i.e., $| f_{a}^{'} | ≫ 1$ away from small neighborhoods of the critical set ${f_{a}^{'} = 0}$ . We obtain, for any $ε > 0$ , a checkable, finite-time criterion on the parameter $a$ for random perturbations of the map $f_{a}$ to exhibit (i) a unique stationary measure and (ii) a positive Lyapunov exponent comparable to $\int_{S^{1}} log | f_{a}^{'} | d x$ . This stands in contrast with the situation for the deterministic dynamics of $f_{a}$ , the chaotic regimes of which are determined by typically uncheckable, infinite-time conditions. Moreover, our finite-time criterion depends on only $k \sim log (ε^{- 1})$ iterates of the deterministic dynamics of $f_{a}$ , which grows quite slowly as $ε \to 0$ .

Accepté le : 2021-04-23
Publié le : 2022-02-25

DOI : 10.4171/aihpc/11

Classification : 37A50, 37C05, 37C40, 37E10, 37H15

@article{AIHPC_2022__39_2_419_0,
     author = {Blumenthal, Alex and Yang, Yun},
     title = {Positive {Lyapunov} exponent for random perturbations of predominantly expanding multimodal circle maps},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {419--455},
     volume = {39},
     number = {2},
     year = {2022},
     doi = {10.4171/aihpc/11},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.4171/aihpc/11/}
}

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Blumenthal, Alex; Yang, Yun. Positive Lyapunov exponent for random perturbations of predominantly expanding multimodal circle maps. Annales de l'I.H.P. Analyse non linéaire, Tome 39 (2022) no. 2, pp. 419-455. doi: 10.4171/aihpc/11

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