Unique ergodicity for random noninvertible maps on an interval
Brofferio, Sara1; Oppelmayer, Hanna2; Szarek, Tomasz3
1Univ. Paris Est Créteil, Univ. Gustave Eiffel, CNRS, LAMA UMR8050, 94010 Creteil (France)
2Universität Innsbruck, Technikerstrasse 13, 6020 Innsbruck (Austria)
3Institut of Mathematics, Polish Academy of Sciences, Abrahama 18, Sopot (Poland)
Annales de l'Institut Fourier, Online first, 28 p.

In this short note, we investigate noninvertible stochastic dynamical systems on the unit interval $[0,1]$. We provide a handy condition for unique ergodicity for systems that are injective in mean. On the other hand, we give concrete examples where unique ergodicity fails.

Dans cette courte note, nous étudions des systèmes dynamiques stochastiques non inversibles de l’intervalle $[0,1]$. Nous proposons une condition maniable pour assurer l’unique ergodicité des systèmes qui sont injectifs en moyenne. D’autre part, nous construisons des exemples concrets où l’unicité ergodique échoue.

Received:
Revised:
Accepted:
Online First:
DOI: 10.5802/aif.3770
Classification: 37E05, 60G10, 37A50, 28D20
Keywords: unique ergodicity, stationary measures, dynamical systems, random interval maps, entropy
Mots-clés : unique ergodicité, mesures stationnaires, systèmes dynamiques, applications aléatoires de l’intervalle, entropie

Brofferio, Sara  1 ; Oppelmayer, Hanna  2 ; Szarek, Tomasz  3

1 Univ. Paris Est Créteil, Univ. Gustave Eiffel, CNRS, LAMA UMR8050, 94010 Creteil (France)
2 Universität Innsbruck, Technikerstrasse 13, 6020 Innsbruck (Austria)
3 Institut of Mathematics, Polish Academy of Sciences, Abrahama 18, Sopot (Poland) 
Brofferio, Sara; Oppelmayer, Hanna; Szarek, Tomasz. Unique ergodicity for random noninvertible maps on an interval. Annales de l'Institut Fourier, Online first, 28 p.
@unpublished{AIF_0__0_0_A59_0,
     author = {Brofferio, Sara and Oppelmayer, Hanna and Szarek, Tomasz},
     title = {Unique ergodicity for random noninvertible maps on an interval},
     journal = {Annales de l'Institut Fourier},
     year = {2026},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     doi = {10.5802/aif.3770},
     language = {en},
     note = {Online first},
}
TY  - UNPB
AU  - Brofferio, Sara
AU  - Oppelmayer, Hanna
AU  - Szarek, Tomasz
TI  - Unique ergodicity for random noninvertible maps on an interval
JO  - Annales de l'Institut Fourier
PY  - 2026
PB  - Association des Annales de l’institut Fourier
N1  - Online first
DO  - 10.5802/aif.3770
LA  - en
ID  - AIF_0__0_0_A59_0
ER  - 
%0 Unpublished Work
%A Brofferio, Sara
%A Oppelmayer, Hanna
%A Szarek, Tomasz
%T Unique ergodicity for random noninvertible maps on an interval
%J Annales de l'Institut Fourier
%D 2026
%V 0
%N 0
%I Association des Annales de l’institut Fourier
%Z Online first
%R 10.5802/aif.3770
%G en
%F AIF_0__0_0_A59_0

[1] Alsedà, Lluís; Misiurewicz, Michał Random interval homeomorphisms, Publ. Mat., Barc., Volume 58 (2014), pp. 15-36 | MR | Zbl | DOI

[2] Alsmeyer, Gerold; Brofferio, Sara; Buraczewski, Dariusz Asymptotically linear iterated function systems on the real line, Ann. Appl. Probab., Volume 33 (2023) no. 1, pp. 161-199 | DOI | MR | Zbl

[3] Avila, Artur; Viana, Marcelo Extremal Lyapunov exponents: an invariance principle and applications, Invent. Math., Volume 181 (2010) no. 1, pp. 115-189 | DOI | MR | Zbl

[4] Baxendale, Peter H. Lyapunov exponents and relative entropy for a stochastic flow of diffeomorphisms, Probab. Theory Relat. Fields, Volume 81 (1989) no. 4, pp. 521-554 | DOI | MR | Zbl

[5] Benoist, Yves; Quint, Jean-François Random walks on reductive groups, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge, 62, Springer, 2016 | MR | Zbl | DOI

[6] Brofferio, Sara; Buraczewski, Dariusz; Szarek, Tomasz On uniqueness of invariant measures for random walks on HOMEO + (ℝ), Ergodic Theory Dyn. Syst., Volume 42 (2022) no. 7, pp. 2207-2238 | DOI | MR | Zbl

[7] Czapla, Dawid On the existence and uniqueness of stationary distributions for some piecewise deterministic Markov processes with state-dependent jump intensity, Results Math., Volume 79 (2024) no. 5, 177, 32 pages | DOI | Zbl | MR

[8] Czudek, Klaudiusz; Szarek, Tomasz Ergodicity and central limit theorem for random interval homeomorphisms, Isr. J. Math., Volume 239 (2020) no. 1, pp. 75-98 | DOI | MR | Zbl

[9] Deroin, Bertrand; Kleptsyn, Victor; Navas, Andrés Sur la dynamique unidimensionnelle en régularité intermédiaire, Acta Math., Volume 199 (2007) no. 2, pp. 199-262 | DOI | MR | Zbl

[10] Deroin, Bertrand; Kleptsyn, Victor; Navas, Andrés; Parwani, K. Symmetric random walks on Homeo + (ℝ), Ann. Probab., Volume 41 (2013) no. 3B, pp. 2066-2089 | DOI | MR | Zbl

[11] Federer, Herbert Geometric measure theory, Classics in Mathematics, 153, Springer, 1996 | DOI | Zbl | MR

[12] Gharaei, Masoumeh; Homburg, Ale Jan Random interval diffeomorphisms, Discrete Contin. Dyn. Syst., Ser. S, Volume 10 (2017) no. 2, pp. 241-272 | DOI | MR | Zbl

[13] Homburg, Ale Jan; Kalle, Charlene; Ruziboev, Marks; Verbitskiy, Evgeny; Zeegers, Benthen Critical intermittency in random interval maps, Commun. Math. Phys., Volume 394 (2022) no. 1, pp. 1-37 | DOI | MR | Zbl

[14] Kloeckner, Benoît R. Optimal transportation and stationary measures for iterated function systems, Math. Proc. Camb. Philos. Soc., Volume 173 (2022) no. 1, pp. 163-187 | DOI | MR | Zbl

[15] Lasota, Andrzej; Mackey, Michael C. Chaos, fractals, and noise, Applied Mathematical Sciences, 97, Springer, 1994 | DOI | MR | Zbl

[16] Ledrappier, F. Positivity of the exponent for stationary sequences of matrices, Lyapunov exponents (Bremen, 1984) (Lecture Notes in Mathematics), Volume 1186, Springer, 1986, pp. 56-73 | DOI | MR | Zbl

[17] Malicet, Dominique Random walks on Homeo(S 1 ), Commun. Math. Phys., Volume 356 (2017) no. 3, pp. 1083-1116 | DOI | MR

[18] Mattila, Pertti Geometry of sets and measures in Euclidean spaces, Cambridge Studies in Advanced Mathematics, 44, Cambridge University Press, 1995 | DOI | MR

[19] Navas, Andrés Groups of circle diffeomorphisms, Chicago Lectures in Mathematics, University of Chicago Press, 2011 | DOI | MR

[20] Nowicki, Tomasz; van Strien, Sebastian Invariant measures exist under a summability condition for unimodal maps, Invent. Math., Volume 105 (1991) no. 1, pp. 123-136 | DOI | MR

Cited by Sources: