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

Blumenthal, Alex; Yang, Yun

doi:10.4171/aihpc/11

Blumenthal, Alex ; Yang, Yun

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

Le texte intégral des articles récents est réservé aux abonnés de la revue. Consultez l'article sur le site de la revue.

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},
     year = {2022},
     volume = {39},
     number = {2},
     doi = {10.4171/aihpc/11},
     language = {en},
     url = {https://www.numdam.org/articles/10.4171/aihpc/11/}
}

TY  - JOUR
AU  - Blumenthal, Alex
AU  - Yang, Yun
TI  - Positive Lyapunov exponent for random perturbations of predominantly expanding multimodal circle maps
JO  - Annales de l'I.H.P. Analyse non linéaire
PY  - 2022
SP  - 419
EP  - 455
VL  - 39
IS  - 2
UR  - https://www.numdam.org/articles/10.4171/aihpc/11/
DO  - 10.4171/aihpc/11
LA  - en
ID  - AIHPC_2022__39_2_419_0
ER  -

%0 Journal Article
%A Blumenthal, Alex
%A Yang, Yun
%T Positive Lyapunov exponent for random perturbations of predominantly expanding multimodal circle maps
%J Annales de l'I.H.P. Analyse non linéaire
%D 2022
%P 419-455
%V 39
%N 2
%U https://www.numdam.org/articles/10.4171/aihpc/11/
%R 10.4171/aihpc/11
%G en
%F AIHPC_2022__39_2_419_0

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

Cité par Sources :