Tokyo Journal of Mathematics

Historic Behaviour for Random Expanding Maps on the Circle


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F. Takens constructed a residual subset of the state space consisting of initial points with historic behaviour for expanding maps on the circle. We prove that this statistical property of expanding maps on the circle is preserved under small random perturbations. The proof is given by establishing a random Markov partition, which follows from a random version of Shub's Theorem on topological conjugacy with the folding maps.

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Tokyo J. Math., Volume 40, Number 1 (2017), 165-184.

First available in Project Euclid: 8 August 2017

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Zentralblatt MATH identifier

Primary: 37C40: Smooth ergodic theory, invariant measures [See also 37Dxx]
Secondary: 37H10: Generation, random and stochastic difference and differential equations [See also 34F05, 34K50, 60H10, 60H15]


NAKANO, Yushi. Historic Behaviour for Random Expanding Maps on the Circle. Tokyo J. Math. 40 (2017), no. 1, 165--184. doi:10.3836/tjm/1502179221.

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  • V. Araújo, Attractors and time averages for random maps, Ann. lnst. H. Poincaré Anal. Non Linéaire 17 (2000), 307–369.
  • V. Baladi, M. Benedicks and V. Maume-Deschamps, Almost sure rates of mixing for i.i.d. unimodal maps, Ann. Sci. École Norm. Sup. 35 (2002), 77–126.
  • L. Barreira and J. Schmeling, Sets of “non-typical” points have full topological entropy and full Hausdorff dimension, Israel J. Math. 116 (2000), 29–70.
  • C. Bonatti, L. Díaz and M. Viana, Dynamics Beyond Uniform Hyperbolicity: A Global Geometric and Probabilistic Perspective, Encyclopaedia of Mathematical Science 102, Springer-Verlag, 2004.
  • I. P. Cornfeld, S. V. Fomin and Y. G. Sinai, Ergodic Theory, 245, Springer Science & Business Media, 2012.
  • F. Hofbauer and G. Keller, Quadratic maps without asymptotic measure, Comm. Math. Phys. 127 (1990), 319–337.
  • A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems, Encyclopedia of Mathematics and its Applications 54, Cambridge Univ. Press, 1995.
  • S. Kiriki, M.-C. Li and T. Soma, Geometric Lorenz flows with historic behavior, Discrete Contin. Dyn. Syst. 36 (2016), 7021–7028.
  • S. Kiriki and T. Soma, Takens' last problem and existence of non-trivial wandering domains, Adv. Math. 306 (2017), 524–588.
  • I. S. Labouriau and A. A. P. Rodrigues, On Takens' last problem: Tangencies and time averages near heteroclinic networks, arXiv preprint arXiv:1606.07017 (2016).
  • D. Ruelle, Historical behaviour in smooth dynamical systems, Global Analysis of Dynamical Systems, eds. H. W. Broer et al., Institute of Physics Publishing, 2001.
  • F. Takens, Orbits with historic behaviour, or non-existence of averages, Nonlinearity 21 (2008), T33–T36.
  • M. Viana, Lecture Notes on Attractors and Physical Measures, IMCA, 1999.
  • P. Walters, An Introduction to Ergodic Theory, 79, Springer Science & Business Media, 2000.