American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Existence of solutions of the equations of electron magnetohydrodynamics in a bounded domain
  • DCDS Home
  • This Issue
  • Next Article
    Pullback attractors for 2D-Navier-Stokes equations with delays in continuous and sub-linear operators
September  2010, 26(3): 1007-1018. doi: 10.3934/dcds.2010.26.1007

Ulam's method for some non-uniformly expanding maps

Rua Murray 1,

1. 

Department of Mathematics and Statistics, University of Canterbury, Private Bag 4800, Christchurch 8140, New Zealand

Received  March 2009 Revised  September 2009 Published  December 2009

Certain dynamical systems on the interval with indifferent fixed points admit invariant probability measures which are absolutely continuous with respect to Lebesgue measure. These maps are often used as a model of intermittent dynamics, and they exhibit sub-exponential decay of correlations (due to the absence of a spectral gap in the underlying transfer operator). This paper concerns a class of these maps which are expanding (with convex branches), but admit an indifferent fixed point with tangency of $O(x^{1+\alpha})$ at $x=0$ ($0<\alpha<1$). The main results show that invariant probability measures can be rigorously approximated by a finite calculation. More precisely: Ulam's method (a sequence of computable finite rank approximations to the transfer operator) exhibits $L^1$ - convergence; and the $n$th approximate invariant density is accurate to at least $O(n^{-(1-\alpha)^2})$. Explicitly given non-uniform Ulam methods can improve this rate to $O(n^{-(1-\alpha)})$.
Keywords: indifferent fixed point - invariant measure approximation - non-uniformly expanding dynamical system - mixing time - polynomial decay of correlations - Ulam's method..
Mathematics Subject Classification: Primary: 37M25; Secondary: 28D0.
Citation: Rua Murray. Ulam's method for some non-uniformly expanding maps. Discrete and Continuous Dynamical Systems, 2010, 26 (3) : 1007-1018. doi: 10.3934/dcds.2010.26.1007
[1]

Yunping Jiang, Yuan-Ling Ye. Convergence speed of a Ruelle operator associated with a non-uniformly expanding conformal dynamical system and a Dini potential. Discrete and Continuous Dynamical Systems, 2018, 38 (9) : 4693-4713. doi: 10.3934/dcds.2018206
[2]

Yuan-Ling Ye. Non-uniformly expanding dynamical systems: Multi-dimension. Discrete and Continuous Dynamical Systems, 2019, 39 (5) : 2511-2553. doi: 10.3934/dcds.2019106
[3]

José F. Alves. Non-uniformly expanding dynamics: Stability from a probabilistic viewpoint. Discrete and Continuous Dynamical Systems, 2001, 7 (2) : 363-375. doi: 10.3934/dcds.2001.7.363
[4]

Christopher Bose, Rua Murray. The exact rate of approximation in Ulam's method. Discrete and Continuous Dynamical Systems, 2001, 7 (1) : 219-235. doi: 10.3934/dcds.2001.7.219
[5]

Xueting Tian, Paulo Varandas. Topological entropy of level sets of empirical measures for non-uniformly expanding maps. Discrete and Continuous Dynamical Systems, 2017, 37 (10) : 5407-5431. doi: 10.3934/dcds.2017235
[6]

José F. Alves. A survey of recent results on some statistical features of non-uniformly expanding maps. Discrete and Continuous Dynamical Systems, 2006, 15 (1) : 1-20. doi: 10.3934/dcds.2006.15.1
[7]

Jose F. Alves; Stefano Luzzatto and Vilton Pinheiro. Markov structures for non-uniformly expanding maps on compact manifolds in arbitrary dimension. Electronic Research Announcements, 2003, 9: 26-31.

[8]

Karla Díaz-Ordaz. Decay of correlations for non-Hölder observables for one-dimensional expanding Lorenz-like maps. Discrete and Continuous Dynamical Systems, 2006, 15 (1) : 159-176. doi: 10.3934/dcds.2006.15.159
[9]

Michiko Yuri. Polynomial decay of correlations for intermittent sofic systems. Discrete and Continuous Dynamical Systems, 2008, 22 (1&2) : 445-464. doi: 10.3934/dcds.2008.22.445
[10]

Nicolai T. A. Haydn, Kasia Wasilewska. Limiting distribution and error terms for the number of visits to balls in non-uniformly hyperbolic dynamical systems. Discrete and Continuous Dynamical Systems, 2016, 36 (5) : 2585-2611. doi: 10.3934/dcds.2016.36.2585
[11]

F. Rodriguez Hertz, M. A. Rodriguez Hertz, A. Tahzibi and R. Ures. A criterion for ergodicity for non-uniformly hyperbolic diffeomorphisms. Electronic Research Announcements, 2007, 14: 74-81. doi: 10.3934/era.2007.14.74
[12]

Jérôme Buzzi, Véronique Maume-Deschamps. Decay of correlations on towers with non-Hölder Jacobian and non-exponential return time. Discrete and Continuous Dynamical Systems, 2005, 12 (4) : 639-656. doi: 10.3934/dcds.2005.12.639
[13]

Mikhail B. Sevryuk. Invariant tori in quasi-periodic non-autonomous dynamical systems via Herman's method. Discrete and Continuous Dynamical Systems, 2007, 18 (2&3) : 569-595. doi: 10.3934/dcds.2007.18.569
[14]

Marco Lenci. Uniformly expanding Markov maps of the real line: Exactness and infinite mixing. Discrete and Continuous Dynamical Systems, 2017, 37 (7) : 3867-3903. doi: 10.3934/dcds.2017163
[15]

Vincent Lynch. Decay of correlations for non-Hölder observables. Discrete and Continuous Dynamical Systems, 2006, 16 (1) : 19-46. doi: 10.3934/dcds.2006.16.19
[16]

Snir Ben Ovadia. Symbolic dynamics for non-uniformly hyperbolic diffeomorphisms of compact smooth manifolds. Journal of Modern Dynamics, 2018, 13: 43-113. doi: 10.3934/jmd.2018013
[17]

Carlos Matheus, Jacob Palis. An estimate on the Hausdorff dimension of stable sets of non-uniformly hyperbolic horseshoes. Discrete and Continuous Dynamical Systems, 2018, 38 (2) : 431-448. doi: 10.3934/dcds.2018020
[18]

Boris Kalinin, Victoria Sadovskaya. Lyapunov exponents of cocycles over non-uniformly hyperbolic systems. Discrete and Continuous Dynamical Systems, 2018, 38 (10) : 5105-5118. doi: 10.3934/dcds.2018224
[19]

Suzete Maria Afonso, Vanessa Ramos, Jaqueline Siqueira. Equilibrium states for non-uniformly hyperbolic systems: Statistical properties and analyticity. Discrete and Continuous Dynamical Systems, 2021, 41 (9) : 4485-4513. doi: 10.3934/dcds.2021045
[20]

Stefano Galatolo, Pietro Peterlongo. Long hitting time, slow decay of correlations and arithmetical properties. Discrete and Continuous Dynamical Systems, 2010, 27 (1) : 185-204. doi: 10.3934/dcds.2010.27.185

2020 Impact Factor: 1.392

Tools

Metrics

  • PDF downloads (97)
  • HTML views (0)
  • Cited by (18)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy