Hostname: page-component-cd9895bd7-jkksz Total loading time: 0 Render date: 2024-12-28T00:57:41.114Z Has data issue: false hasContentIssue false
  • English
  • Français

Rigorous computation of invariant measures and fractal dimension for maps with contracting fibers: 2D Lorenz-like maps

Published online by Cambridge University Press:  13 April 2015

STEFANO GALATOLO  and
ISAIA NISOLI
STEFANO GALATOLO
Affiliation:
Dipartimento di Matematica, Universita di Pisa, Via Buonarroti 1, Pisa, Italy email galatolo@dm.unipi.it
ISAIA NISOLI
Affiliation:
Instituto de Matemática, UFRJ Av. Athos da Silveira Ramos 149, Centro de Tecnologia, Bloco C Cidade Universitária, Ilha do Fundão, Caixa Postal 68530, 21941-909 Rio de Janeiro, RJ, Brasil email nisoli@im.ufrj.br
Get access Rights & Permissions [Opens in a new window]

Abstract

We consider a class of maps from the unit square to itself preserving a contracting foliation and inducing a one-dimensional map having an absolutely continuous invariant measure. We show how the physical measure of those systems can be rigorously approximated with an explicitly given bound on the error with respect to the Wasserstein distance. We present a rigorous implementation of our algorithm using interval arithmetics, and the result of the computation on a non-trivial example of a Lorenz-like two-dimensional map and its attractor, obtaining a statement on its local dimension.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 36 , Issue 6 , September 2016 , pp. 1865 - 1891
Copyright
© Cambridge University Press, 2015 

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

Araujio, V., Galatolo, S. and Pacifico, M. J.. Decay of correlations for maps with uniformly contracting fibers and logarithm law for singular hyperbolic attractors. Math. Z. 276(3–4) (2014), 1001–1048.CrossRefGoogle Scholar
Araujio, V. and Pacifico, M. J.. Three-Dimensional Flows (A Series Of Modern Surveys in Mathematics) . Springer, Berlin, 2010.Google Scholar
Bahsoun, W. and Bose, C.. Invariant densities and escape rates: rigorous and computable estimation in the L ∞ norm. Nonlinear Anal. 74 (2011), 4481–4495.Google Scholar
Dellnitz, M. and Junge, O.. Set oriented numerical methods for dynamical systems. Handbook of Dynamical Systems. Vol. 2. World Scientific, Singapore, 2002.Google Scholar
Ding, J. and Zhou, A.. The projection method for computing multidimensional absolutely continuous invariant measures. J. Stat. Phys. 77 (1994), 899–908.Google Scholar
Froyland, G.. Finite approximation of Sinai-Bowen-Ruelle measures of Anosov systems in two dimensions. Random Comput. Dynam. 3(4) (1995), 251–264.Google Scholar
Froyland, G.. Extracting dynamical behaviour via Markov models. Nonlinear Dynamics and Statistics: Proceedings Newton Institute, Cambridge, 1998. Ed. Mees, A.. Birkhäuser, Boston, MA, 2001, pp. 283–324.Google Scholar
Galatolo, S., Hoyrup, M. and Rojas, C.. A constructive Borel–Cantelli lemma constructing orbits with required statistical properties. Theoret. Comput. Sci. 410 (2009), 2207–2222.CrossRefGoogle Scholar
Galatolo, S., Hoyrup, M. and Rojas, C.. Dynamics and abstract computability: computing invariant measures. Discrete Contin. Dyn. Syst. 29(1) (2011), 193–212.Google Scholar
Galatolo, S. and Nisoli, I.. An elementary approach to rigorous approximation of invariant measures. SIAM J. Appl. Dyn. Syst. 13(2) (2014), 958–985.Google Scholar
Galatolo, S., Nisoli, I. and Saussol, B.. An elementary way to rigorously estimate convergence to equilibrium and escape rates. Preprint, 2014, arXiv:1404.7113.Google Scholar
Galatolo, S. and Pacifico, M. J.. Lorenz-like flows: exponential decay of correlations for the Poincaré map, logarithm law, quantitative recurrence. Ergod. Th. & Dynam. Sys. 30 (2010), 703–1737.Google Scholar
Keller, G.. Generalized bounded variation and applications to piecewise monotonic transformations. Z. Wahrsch. Verw. Gebiete 69(3) (1985), 461–478.Google Scholar
Ippei, O.. Computer-assisted verification method for invariant densities and rates of decay of correlations. SIAM J. Appl. Dyn. Syst. 10(2) (2011), 788–816.CrossRefGoogle Scholar
Lasota, A. and Yorke, J.. On the existence of invariant measures for piecewise monotonic transformations. Trans. Amer. Math. Soc. 186 (1973), 481–488.Google Scholar
Liverani, C.. Rigorous numerical investigations of the statistical properties of piecewise expanding maps–A feasibility study. Nonlinearity 14 (2001), 463–490.Google Scholar
Liverani, C.. Invariant measures and their properties. A functional analytic point of view. Dynamical Systems Part II: Topological Geometrical and Ergodic Properties of Dynamics. Centro di Ricerca Matematica ‘Ennio De Giorgi’: Proceedings. Scuola Normale Superiore, Pisa, 2004.Google Scholar
Gouezel, S. and Liverani, C.. Banach spaces adapted to Anosov systems. Ergod. Th. & Dynam. Sys. 26 (2006), 189–217.Google Scholar
Keane, M., Murray, R. and Young, L. S.. Computing invariant measures for expanding circle maps. Nonlinearity 11 (1998), 27–46.Google Scholar
Pollicott, M. and Jenkinson, O.. Computing invariant densities and metric entropy. Comm. Math. Phys. 211 (2000), 687–703.Google Scholar
Steinberger, T.. Local dimension of ergodic measures for two-dimensional Lorenz transformations. Ergod. Th. & Dynam. Sys. 20(3) (2000), 911–923.Google Scholar
Young, L.-S.. What are SRB measures, and which dynamical systems have them? J. Stat. Phys. 108 (2002), 733–754.Google Scholar