Hostname: page-component-cd9895bd7-gbm5v Total loading time: 0 Render date: 2024-12-27T05:44:17.429Z Has data issue: false hasContentIssue false
  • English
  • Français

Approximating integrals with respect to stationary probability measures of iterated function systems

Part of: Low-dimensional dynamical systems Approximation methods and numerical treatment of dynamical systems Dynamical systems with hyperbolic behavior

Published online by Cambridge University Press:  08 June 2020

ITALO CIPRIANO  and
NATALIA JURGA
ITALO CIPRIANO
Affiliation:
Facultad de Matemáticas, Pontificia Universidad Católica de Chile (PUC), Avenida Vicuña Mackenna 4860, Santiago, Chile (e-mail: icipriano@gmail.com)
NATALIA JURGA
Affiliation:
Mathematical Institute, St Andrews, KY16 9SS, UK (e-mail: naj1@st-andrews.ac.uk)
Get access Rights & Permissions [Opens in a new window]

Abstract

We study fast approximation of integrals with respect to stationary probability measures associated to iterated function systems on the unit interval. We provide an algorithm for approximating the integrals under certain conditions on the iterated function system and on the function that is being integrated. We apply this technique to estimate Hausdorff moments, Wasserstein distances and Lyapunov exponents of stationary probability measures.

Keywords

low-dimensional dynamicsiterated function systemstransfer operatorstationary measures

MSC classification

Primary: 37M25: Computational methods for ergodic theory (approximation of invariant measures, computation of Lyapunov exponents, entropy) 37E05: Maps of the interval (piecewise continuous, continuous, smooth) 37D20: Uniformly hyperbolic systems (expanding, Anosov, Axiom A, etc.)
Secondary: 37D35: Thermodynamic formalism, variational principles, equilibrium states
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 41 , Issue 8 , August 2021 , pp. 2294 - 2310
Check for updates
Copyright
© The Author(s) 2020. Published by Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Akhiezer, N. I.. The Classical Moment Problem and Some Related Questions in Analysis. Hafner, New York, 1965. Translated by Kemmer, N..Google Scholar
Bandtlow, O. F. and Jenkinson, O.. On the Ruelle eigenvalue sequence. Ergod. Th. & Dynam. Sys. 28(6) (2008), 1701–1711.Google Scholar
Bandtlow, O. F. and Jenkinson, O.. Explicit eigenvalue estimates for transfer operators acting on spaces of holomorphic functions. Adv. Math. 218(3) (2008), 902–925.Google Scholar
Blank, M.. Perron–Frobenius spectrum for random maps and its approximation. Mosc. Math. J. 1(3) (2001), 315–344, 470.Google Scholar
Cipriano, I.. The Wasserstein distance between stationary measures. Preprint, 2018, arXiv:1611.00092v3.Google Scholar
Cipriano, I. and Pollicott, M.. Stationary measures associated to analytic iterated function schemes. Math. Nachr. 291(7) (2018), 1049–1054.Google Scholar
Diaconis, P.. Application of the method of moments in probability and statistics. Moments in Mathematics (San Antonio, TX, 1987) (Proceedings of Symposia in Applied Mathematics, 37). American Mathematical Society, Providence, RI, 1987, pp. 125–142.Google Scholar
Falconer, K.. Fractal Geometry: Mathematical Foundations and Applications, 3rd edn. John Wiley & Sons, Chichester, 2014.Google Scholar
Fan, A. H. and Lau, K.-S.. Iterated function system and Ruelle operator. J. Math. Anal. Appl. 231 (1999), 319–344.Google Scholar
Fraser, J.. First and second moments for self-couplings and Wasserstein distances. Math. Nachr. 288 (2015), 2028–2041.Google Scholar
Froyland, G.. Ulam’s method for random interval maps. Nonlinearity 12 (1999), 1029–1052.Google Scholar
Galatolo, S., Monge, M. and Nisoli, I.. Rigorous approximation of stationary measures and convergence to equilibrium for iterated function systems. J. Phys. A 49 (2016), 274001.Google Scholar
Hennion, H. and Hervé, L.. Limit Theorems for Markov Chains and Stochastic Properties of Dynamical Systems by Quasi-Compactness. Vol. 1766. Springer Science and Business Media, Berlin, 2001.Google Scholar
Hutchinson, J.. Fractals and self similarity. Indiana Univ. Math. J. 30 (1981), 713–747.Google Scholar
Jenkinson, O. and Pollicott, M.. A dynamical approach to accelerating numerical integration with equidistributed points. Proc. Steklov Inst. Math. 256(1) (2007), 275–289.Google Scholar
Jenkinson, O. and Pollicott, M.. Rigorous effective bounds on the Hausdorff dimension of continued fraction Cantor sets: a hundred decimal digits for the dimension of E2. Adv. Math. 325 (2018), 87–115.Google Scholar
Jurga, N. and Morris, I.. Effective estimates on the top Lyapunov exponent for random matrix products. Nonlinearity 32(11) (2019), 4117–4146.Google Scholar
Kato, T.. Perturbation Theory for Linear Operators (Classics in Mathematics). Reprint of the 1980 edn. Springer, Berlin, 1995.Google Scholar
Öberg, A.. Algorithms for approximation of invariant measures for IFS. Manuscripta Math. 116 (2005), 31–55.Google Scholar
Ruelle, D.. Zeta-functions for expanding maps and Anosov flows. Invent. Math. 34(3) (1976), 231–242.Google Scholar
Simon, B.. Trace Ideals and their Applications. Vol. 120. American Mathematical Society, Providence, RI, 2010.Google Scholar
Strichartz, R., Taylor, A. and Zhang, T.. Densities of self-similar measures on the line. Exp. Math. 4 (995), 101–128.Google Scholar