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Rigorous pointwise approximations for invariant densities of non-uniformly expanding maps

Published online by Cambridge University Press:  10 January 2014

WAEL BAHSOUN ,
CHRISTOPHER BOSE  and
YUEJIAO DUAN
WAEL BAHSOUN
Affiliation:
Department of Mathematical Sciences, Loughborough University, Loughborough, Leicestershire, LE11 3TU, UK email [email protected]@lboro.ac.uk
CHRISTOPHER BOSE
Affiliation:
Department of Mathematics and Statistics, University of Victoria, PO Box 3045 STN CSC, Victoria, BC, V8W 3R4, Canada email [email protected]
YUEJIAO DUAN
Affiliation:
Department of Mathematical Sciences, Loughborough University, Loughborough, Leicestershire, LE11 3TU, UK email [email protected]@lboro.ac.uk
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Abstract

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We use an Ulam-type discretization scheme to provide pointwise approximations for invariant densities of interval maps with a neutral fixed point. We prove that the approximate invariant density converges pointwise to the true density at a rate ${C}^{\ast } \cdot (\ln m)/ m$, where ${C}^{\ast } $ is a computable fixed constant and ${m}^{- 1} $ is the mesh size of the discretization.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 35 , Issue 4 , June 2015 , pp. 1028 - 1044
Copyright
© Cambridge University Press, 2014 

References

Aytac, H., Freitas, J. and Vaienti, S.. Laws of rare events for deterministic and random dynamical systems. Preprint, 2012, arxiv.org/pdf/1207.5188.Google Scholar
Bahsoun, W. and Bose, C.. Invariant densities and escape rates: rigorous and computable approximations in the ${L}^{\infty } $-norm. Nonlinear Anal. 74 (2011), 4481–4495.CrossRefGoogle Scholar
Bahsoun, W. and Vaienti, S.. Metastability of certain intermittent maps. Nonlinearity 25 (1) (2012), 107–124.CrossRefGoogle Scholar
Baladi, V.. Positive Transfer Operators and Decay of Correlations (Advanced Series in Nonlinear Dynamics, 16). World Science Publications, River Edge, NJ, 2000.CrossRefGoogle Scholar
Blank, M.. Finite rank approximations of expanding maps with neutral singularities. Discrete Contin. Dyn. Syst. 21 (3) (2008), 749–762.CrossRefGoogle Scholar
Blank, M., Keller, G. and Liverani, C.. Ruelle–Perron–Frobenius spectrum for Anosov maps. Nonlinearity 15 (6) (2002), 1905–1973.CrossRefGoogle Scholar
Bose, C.. Invariant measures and equilibrium states for piecewise ${C}^{1+ \alpha } $ endomorphisms of the unit interval. Trans. Amer. Math. Soc. 315 (1) (1989), 105–125.Google Scholar
Boyarsky, A. and Góra, P.. Laws of Chaos: Invariant Measures and Dynamical Systems in one Dimension. Birkhäuser, Boston, 1997.CrossRefGoogle Scholar
Ding, J. and Li, T. A.. Convergence rate analysis for Markov approximations to a class of Frobenius–Perron operators. Nonlinear Anal. 31 (1998), 765–777.CrossRefGoogle Scholar
Froyland, G.. Finite approximation of Sinai–Bowen–Ruelle measures for Anosov systems in two dimensions. Random Comput. Dynam. 3 (4) (1995), 251–263.Google Scholar
Froyland, G.. On Ulam approximation of the isolated spectrum and eigenfunctions of hyperbolic maps. Discrete Contin. Dyn. Syst. 17 (3) (2007), 671–689.CrossRefGoogle Scholar
Galatolo, S. and Nisoli, I.. An elementary approach to rigorous approximation of invariant measures. Preprint, 2011, arXiv:1109.2342.Google Scholar
Holland, M., Nicol, M. and Török, A.. Extreme value theory for non-uniformly expanding dynamical systems. Trans. Amer. Math. Soc. 364 (2) (2012), 661–688.CrossRefGoogle Scholar
Hu, H.. Decay of correlations for piecewise smooth maps with indifferent fixed points. Ergod. Th. & Dynam. Sys. 24 (2) (2004), 495–524.CrossRefGoogle Scholar
Keller, G.. Rare events, exponential hitting times and extremal indices via spectral perturbation. Dyn. Syst. 27 (1) (2012), 11–27.CrossRefGoogle Scholar
Keller, G. and Liverani, C.. Stability of the spectrum for transfer operators. Ann. Sc. Norm. Super. Pisa Cl. Sci. (4) 28 (1) (1999), 141–152.Google Scholar
Lancaster, P. and Tismenetsky, M.. The Theory of Matrices. Academic Press, Orlando, FL, 1985.Google Scholar
Liverani, C.. Rigorous numerical investigation of the statistical properties of piecewise expanding maps. A feasibility study. Nonlinearity 14 (3) (2001), 463–490.CrossRefGoogle Scholar
Liverani, C., Saussol, B. and Vaienti, S.. A probabilistic approach to intermittency. Ergod. Th. & Dynam. Sys. 19 (1999), 671–685.CrossRefGoogle Scholar
Murray, R.. Existence, mixing and approximation of invariant densities for expanding maps on ${R}^{r} $. Nonlinear Anal. 45 (1) (2001), 37–72.CrossRefGoogle Scholar
Murray, R.. Ulam’s method for some non-uniformly expanding maps. Discrete. Contin. Dyn. Syst. 26 (3) (2010), 1007–1018.CrossRefGoogle Scholar
Pianigiani, G.. First return map and invariant measures. Israel J. Math. 35 (1980), 32–48.CrossRefGoogle Scholar
Ulam, S. M.. A Collection of Mathematical Problems (Interscience Tracts in Pure and Applied Mathematics, 8). Interscience, New York, 1960.Google Scholar
Young, L.-S.. Recurrence times and rates of mixing. Israel J. Math. 110 (1999), 153–188.CrossRefGoogle Scholar