American Institute of Mathematical Sciences

Advanced
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

Romain Aimino 1, , Huyi Hu 2, , Matthew Nicol 3, , Andrei Török 3, and  Sandro Vaienti 1,

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.
Keywords: non-stationary dynamics, sequential systems, distortion, loss of memory, neutral fixed point, Intermittency, polynomial decorrelation..
Mathematics Subject Classification: 37E05, 37A25, 37H99, 37M9.
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, 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). 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), 1543-1554. arXiv:1305.6588. 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, Contemp. Math., Amer. Math. Soc., Providence, RI, 430 (2007), 89-121. doi: 10.1090/conm/430/08253.  Google Scholar

[5]

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

[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.  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), 141-161. 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), 495-524. 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), 671-685. 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), 463-475. doi: 10.4310/MRL.2009.v16.n3.a7.  Google Scholar

[11]

O. Sarig, Subexponential decay of correlations, Invent. Math., 150 (2002), 629-653. 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, An International Journal, 28 (2013), 423-452. doi: 10.1080/14689367.2013.806733.  Google Scholar

[13]

M. Stenlund, Non-stationary compositions of Anosov diffeomorphisms, Nonlinearity, 24 (2011), 2991-3018. 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), 909-955. doi: 10.1007/s00220-013-1746-6.  Google Scholar

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). 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), 1543-1554. arXiv:1305.6588. 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, Contemp. Math., Amer. Math. Soc., Providence, RI, 430 (2007), 89-121. doi: 10.1090/conm/430/08253.  Google Scholar

[5]

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

[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.  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), 141-161. 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), 495-524. 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), 671-685. 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), 463-475. doi: 10.4310/MRL.2009.v16.n3.a7.  Google Scholar

[11]

O. Sarig, Subexponential decay of correlations, Invent. Math., 150 (2002), 629-653. 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, An International Journal, 28 (2013), 423-452. doi: 10.1080/14689367.2013.806733.  Google Scholar

[13]

M. Stenlund, Non-stationary compositions of Anosov diffeomorphisms, Nonlinearity, 24 (2011), 2991-3018. 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), 909-955. doi: 10.1007/s00220-013-1746-6.  Google Scholar

[1]

Yaofeng Su. Almost surely invariance principle for non-stationary and random intermittent dynamical systems. Discrete & Continuous Dynamical Systems, 2019, 39 (11) : 6585-6597. doi: 10.3934/dcds.2019286
[2]

Fredi Tröltzsch, Alberto Valli. Optimal voltage control of non-stationary eddy current problems. Mathematical Control & Related Fields, 2018, 8 (1) : 35-56. doi: 10.3934/mcrf.2018002
[3]

Hi Jun Choe, Do Wan Kim, Yongsik Kim. Meshfree method for the non-stationary incompressible Navier-Stokes equations. Discrete & Continuous Dynamical Systems - B, 2006, 6 (1) : 17-39. doi: 10.3934/dcdsb.2006.6.17
[4]

Anushaya Mohapatra, William Ott. Memory loss for nonequilibrium open dynamical systems. Discrete & Continuous Dynamical Systems, 2014, 34 (9) : 3747-3759. doi: 10.3934/dcds.2014.34.3747
[5]

Kaïs Ammari, Thomas Duyckaerts, Armen Shirikyan. Local feedback stabilisation to a non-stationary solution for a damped non-linear wave equation. Mathematical Control & Related Fields, 2016, 6 (1) : 1-25. doi: 10.3934/mcrf.2016.6.1
[6]

Yinnian He, Yanping Lin, Weiwei Sun. Stabilized finite element method for the non-stationary Navier-Stokes problem. Discrete & Continuous Dynamical Systems - B, 2006, 6 (1) : 41-68. doi: 10.3934/dcdsb.2006.6.41
[7]

Sherif I. Ammar, Alexander Zeifman, Yacov Satin, Ksenia Kiseleva, Victor Korolev. On limiting characteristics for a non-stationary two-processor heterogeneous system with catastrophes, server failures and repairs. Journal of Industrial & Management Optimization, 2021, 17 (3) : 1057-1068. doi: 10.3934/jimo.2020011
[8]

Zhendong Luo. A reduced-order SMFVE extrapolation algorithm based on POD technique and CN method for the non-stationary Navier-Stokes equations. Discrete & Continuous Dynamical Systems - B, 2015, 20 (4) : 1189-1212. doi: 10.3934/dcdsb.2015.20.1189
[9]

Chunyou Sun, Daomin Cao, Jinqiao Duan. Non-autonomous wave dynamics with memory --- asymptotic regularity and uniform attractor. Discrete & Continuous Dynamical Systems - B, 2008, 9 (3&4, May) : 743-761. doi: 10.3934/dcdsb.2008.9.743
[10]

Tiphaine Jézéquel, Patrick Bernard, Eric Lombardi. Homoclinic orbits with many loops near a $0^2 i\omega$ resonant fixed point of Hamiltonian systems. Discrete & Continuous Dynamical Systems, 2016, 36 (6) : 3153-3225. doi: 10.3934/dcds.2016.36.3153
[11]

Nicholas Long. Fixed point shifts of inert involutions. Discrete & Continuous Dynamical Systems, 2009, 25 (4) : 1297-1317. doi: 10.3934/dcds.2009.25.1297
[12]

Rafael Obaya, Víctor M. Villarragut. Direct exponential ordering for neutral compartmental systems with non-autonomous $\mathbf{D}$-operator. Discrete & Continuous Dynamical Systems - B, 2013, 18 (1) : 185-207. doi: 10.3934/dcdsb.2013.18.185
[13]

Yakov Krasnov, Alexander Kononovich, Grigory Osharovich. On a structure of the fixed point set of homogeneous maps. Discrete & Continuous Dynamical Systems - S, 2013, 6 (4) : 1017-1027. doi: 10.3934/dcdss.2013.6.1017
[14]

Jorge Groisman. Expansive and fixed point free homeomorphisms of the plane. Discrete & Continuous Dynamical Systems, 2012, 32 (5) : 1709-1721. doi: 10.3934/dcds.2012.32.1709
[15]

Yong Ji, Ercai Chen, Yunping Wang, Cao Zhao. Bowen entropy for fixed-point free flows. Discrete & Continuous Dynamical Systems, 2019, 39 (11) : 6231-6239. doi: 10.3934/dcds.2019271
[16]

Shui-Hung Hou. On an application of fixed point theorem to nonlinear inclusions. Conference Publications, 2011, 2011 (Special) : 692-697. doi: 10.3934/proc.2011.2011.692
[17]

Luis Hernández-Corbato, Francisco R. Ruiz del Portal. Fixed point indices of planar continuous maps. Discrete & Continuous Dynamical Systems, 2015, 35 (7) : 2979-2995. doi: 10.3934/dcds.2015.35.2979
[18]

Antonio Garcia. Transition tori near an elliptic-fixed point. Discrete & Continuous Dynamical Systems, 2000, 6 (2) : 381-392. doi: 10.3934/dcds.2000.6.381
[19]

S. Gatti, Elena Sartori. Well-posedness results for phase field systems with memory effects in the order parameter dynamics. Discrete & Continuous Dynamical Systems, 2003, 9 (3) : 705-726. doi: 10.3934/dcds.2003.9.705
[20]

Tomás Caraballo Garrido, Oleksiy V. Kapustyan, Pavlo O. Kasyanov, José Valero, Michael Zgurovsky. Preface to the special issue "Dynamics and control in distributed systems: Dedicated to the memory of Valery S. Melnik (1952-2007)". Discrete & Continuous Dynamical Systems - B, 2019, 24 (3) : i-v. doi: 10.3934/dcdsb.20193i

2020 Impact Factor: 1.392

Tools

Metrics

  • PDF downloads (110)
  • HTML views (0)
  • Cited by (17)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences