# American Institute of Mathematical Sciences

March  2015, 35(3): 793-806. doi: 10.3934/dcds.2015.35.793

## Polynomial loss of memory for maps of the interval with a neutral fixed point

 1 Aix Marseille Université, CNRS, CPT, UMR 7332, 13288 Marseille, France, France 2 Department of Mathematics, Michigan State University, East Lansing, MI 48824 3 Department of Mathematics, University of Houston, Houston, TX 77204-3008

Received  February 2014 Revised  June 2014 Published  October 2014

We give an example of a sequential dynamical system consisting of intermittent-type maps which exhibits loss of memory with a polynomial rate of decay. A uniform bound holds for the upper rate of memory loss. The maps may be chosen in any sequence, and the bound holds for all compositions.
Citation: Romain Aimino, Huyi Hu, Matthew Nicol, Andrei Török, Sandro Vaienti. Polynomial loss of memory for maps of the interval with a neutral fixed point. Discrete & Continuous Dynamical Systems - A, 2015, 35 (3) : 793-806. doi: 10.3934/dcds.2015.35.793
##### References:
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##### References:
 [1] R. Aimino, Vitesse de mélange et théorèmes limites pour les systèmes dynamiques aléatoires et non-autonomes,, Ph. D. Thesis, (2014).   Google Scholar [2] R. Aimino and J. Rousseau, Concentration inequalities for sequential dynamical systems of the unit interval,, preprint., ().   Google Scholar [3] W. Bahsoun, Ch. Bose and Y. Duan, Decay of correlation for random intermittent maps,, Nonlinearity, 27 (2014), 1543.  doi: 10.1088/0951-7715/27/7/1543.  Google Scholar [4] J.-P. Conze and A. Raugi, Limit theorems for sequential expanding dynamical systems on [0, 1],, Ergodic theory and related fields, 430 (2007), 89.  doi: 10.1090/conm/430/08253.  Google Scholar [5] W. de Melo, S. van Strien, One-dimensional Dynamics,, Springer, (1993).   Google Scholar [6] S. Gouëzel, Central limit theorem and stable laws for intermittent maps,, Probab. Theory Relat. Fields, 128 (2004), 82.  doi: 10.1007/s00440-003-0300-4.  Google Scholar [7] C. Gupta, W. Ott and A. Török, Memory loss for time-dependent piecewise expanding systems in higher dimension,, Mathematical Research Letters, 20 (2013), 141.  doi: 10.4310/MRL.2013.v20.n1.a12.  Google Scholar [8] H. Hu, Decay of correlations for piecewise smooth maps with indifferent fixed points,, Ergodic Theory and Dynamical Systems, 24 (2004), 495.  doi: 10.1017/S0143385703000671.  Google Scholar [9] C. Liverani, B. Saussol and S. Vaienti, A probabilistic approach to intermittency,, Ergodic theory and dynamical systems, 19 (1999), 671.  doi: 10.1017/S0143385799133856.  Google Scholar [10] W. Ott, M. Stenlund and L.-S. Young, Memory loss for time-dependent dynamical systems,, Math. Res. Lett., 16 (2009), 463.  doi: 10.4310/MRL.2009.v16.n3.a7.  Google Scholar [11] O. Sarig, Subexponential decay of correlations,, Invent. Math., 150 (2002), 629.  doi: 10.1007/s00222-002-0248-5.  Google Scholar [12] W. Shen and S. Van Strien, On stochastic stability of expanding circle maps with neutral fixed points,, Dynamical Systems, 28 (2013), 423.  doi: 10.1080/14689367.2013.806733.  Google Scholar [13] M. Stenlund, Non-stationary compositions of Anosov diffeomorphisms,, Nonlinearity, 24 (2011), 2991.  doi: 10.1088/0951-7715/24/10/016.  Google Scholar [14] M. Stenlund, L-S. Young and H. Zhang, Dispersing billiards with moving scatterers,, Comm. Math. Phys., 322 (2013), 909.  doi: 10.1007/s00220-013-1746-6.  Google Scholar
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