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 and Continuous Dynamical Systems, 2015, 35 (3) : 793-806. doi: 10.3934/dcds.2015.35.793
References:
[1]

R. Aimino, Vitesse de mélange et théorèmes limites pour les systèmes dynamiques aléatoires et non-autonomes, Ph. D. Thesis, Université du Sud Toulon Var, (2014).

[2]

R. Aimino and J. Rousseau, Concentration inequalities for sequential dynamical systems of the unit interval, preprint.

[3]

W. Bahsoun, Ch. Bose and Y. Duan, Decay of correlation for random intermittent maps, Nonlinearity, 27 (2014), 1543-1554. arXiv:1305.6588. doi: 10.1088/0951-7715/27/7/1543.

[4]

J.-P. Conze and A. Raugi, Limit theorems for sequential expanding dynamical systems on [0, 1], Ergodic theory and related fields, Contemp. Math., Amer. Math. Soc., Providence, RI, 430 (2007), 89-121. doi: 10.1090/conm/430/08253.

[5]

W. de Melo, S. van Strien, One-dimensional Dynamics, Springer, Berlin, 1993.

[6]

S. Gouëzel, Central limit theorem and stable laws for intermittent maps, Probab. Theory Relat. Fields, 128 (2004), 82-122, doi: 10.1007/s00440-003-0300-4.

[7]

C. Gupta, W. Ott and A. Török, Memory loss for time-dependent piecewise expanding systems in higher dimension, Mathematical Research Letters, 20 (2013), 141-161. doi: 10.4310/MRL.2013.v20.n1.a12.

[8]

H. Hu, Decay of correlations for piecewise smooth maps with indifferent fixed points, Ergodic Theory and Dynamical Systems, 24 (2004), 495-524. doi: 10.1017/S0143385703000671.

[9]

C. Liverani, B. Saussol and S. Vaienti, A probabilistic approach to intermittency, Ergodic theory and dynamical systems, 19 (1999), 671-685. doi: 10.1017/S0143385799133856.

[10]

W. Ott, M. Stenlund and L.-S. Young, Memory loss for time-dependent dynamical systems, Math. Res. Lett., 16 (2009), 463-475. doi: 10.4310/MRL.2009.v16.n3.a7.

[11]

O. Sarig, Subexponential decay of correlations, Invent. Math., 150 (2002), 629-653. doi: 10.1007/s00222-002-0248-5.

[12]

W. Shen and S. Van Strien, On stochastic stability of expanding circle maps with neutral fixed points, Dynamical Systems, An International Journal, 28 (2013), 423-452. doi: 10.1080/14689367.2013.806733.

[13]

M. Stenlund, Non-stationary compositions of Anosov diffeomorphisms, Nonlinearity, 24 (2011), 2991-3018. doi: 10.1088/0951-7715/24/10/016.

[14]

M. Stenlund, L-S. Young and H. Zhang, Dispersing billiards with moving scatterers, Comm. Math. Phys., 322 (2013), 909-955. doi: 10.1007/s00220-013-1746-6.

show all references

References:
[1]

R. Aimino, Vitesse de mélange et théorèmes limites pour les systèmes dynamiques aléatoires et non-autonomes, Ph. D. Thesis, Université du Sud Toulon Var, (2014).

[2]

R. Aimino and J. Rousseau, Concentration inequalities for sequential dynamical systems of the unit interval, preprint.

[3]

W. Bahsoun, Ch. Bose and Y. Duan, Decay of correlation for random intermittent maps, Nonlinearity, 27 (2014), 1543-1554. arXiv:1305.6588. doi: 10.1088/0951-7715/27/7/1543.

[4]

J.-P. Conze and A. Raugi, Limit theorems for sequential expanding dynamical systems on [0, 1], Ergodic theory and related fields, Contemp. Math., Amer. Math. Soc., Providence, RI, 430 (2007), 89-121. doi: 10.1090/conm/430/08253.

[5]

W. de Melo, S. van Strien, One-dimensional Dynamics, Springer, Berlin, 1993.

[6]

S. Gouëzel, Central limit theorem and stable laws for intermittent maps, Probab. Theory Relat. Fields, 128 (2004), 82-122, doi: 10.1007/s00440-003-0300-4.

[7]

C. Gupta, W. Ott and A. Török, Memory loss for time-dependent piecewise expanding systems in higher dimension, Mathematical Research Letters, 20 (2013), 141-161. doi: 10.4310/MRL.2013.v20.n1.a12.

[8]

H. Hu, Decay of correlations for piecewise smooth maps with indifferent fixed points, Ergodic Theory and Dynamical Systems, 24 (2004), 495-524. doi: 10.1017/S0143385703000671.

[9]

C. Liverani, B. Saussol and S. Vaienti, A probabilistic approach to intermittency, Ergodic theory and dynamical systems, 19 (1999), 671-685. doi: 10.1017/S0143385799133856.

[10]

W. Ott, M. Stenlund and L.-S. Young, Memory loss for time-dependent dynamical systems, Math. Res. Lett., 16 (2009), 463-475. doi: 10.4310/MRL.2009.v16.n3.a7.

[11]

O. Sarig, Subexponential decay of correlations, Invent. Math., 150 (2002), 629-653. doi: 10.1007/s00222-002-0248-5.

[12]

W. Shen and S. Van Strien, On stochastic stability of expanding circle maps with neutral fixed points, Dynamical Systems, An International Journal, 28 (2013), 423-452. doi: 10.1080/14689367.2013.806733.

[13]

M. Stenlund, Non-stationary compositions of Anosov diffeomorphisms, Nonlinearity, 24 (2011), 2991-3018. doi: 10.1088/0951-7715/24/10/016.

[14]

M. Stenlund, L-S. Young and H. Zhang, Dispersing billiards with moving scatterers, Comm. Math. Phys., 322 (2013), 909-955. doi: 10.1007/s00220-013-1746-6.

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