# American Institute of Mathematical Sciences

November  2019, 39(11): 6585-6597. doi: 10.3934/dcds.2019286

## Almost surely invariance principle for non-stationary and random intermittent dynamical systems

 Department of Mathematics, University of Houston, Houston, Texas 77204-3008, USA

Received  January 2019 Revised  May 2019 Published  August 2019

We establish almost sure invariance principles (ASIP), a strong form of approximation by Brownian motion, for non-stationary time series arising as observations on sequential maps possessing an indifferent fixed point. These transformations are obtained by perturbing the slope in the Pomeau-Manneville map. Quenched ASIP for random compositions of these maps is also obtained.

Citation: Yaofeng Su. Almost surely invariance principle for non-stationary and random intermittent dynamical systems. Discrete and Continuous Dynamical Systems, 2019, 39 (11) : 6585-6597. doi: 10.3934/dcds.2019286
##### References:
 [1] R. Aimino, H. Y. Hu, M. Nicol, A. Török and S. Vaienti, Polynomial loss of memory for maps of the interval with a neutral fixed point, Discrete Contin. Dyn. Syst., 35 (2015), 793-806.  doi: 10.3934/dcds.2015.35.793. [2] C. Cuny and F. Merlevède, Strong invariance principles with rate for "reverse" martingale differences and applications, J. Theoret. Probab., 28 (2015), 137-183.  doi: 10.1007/s10959-013-0506-z. [3] D. Dragičević, G. Froyland, C. González-Tokman and S. Vaienti, Almost sure invariance principle for random piecewise expanding maps, Nonlinearity, 31 (2018), 2252-2280.  doi: 10.1088/1361-6544/aaaf4b. [4] N. Haydn, M. Nicol, A. Török and S. Vaienti, Almost sure invariance principle for sequential and non-stationary dynamical systems, Trans. Amer. Math. Soc., 369 (2017), 5293-5316.  doi: 10.1090/tran/6812. [5] O. Hella and J. Leppänen, Central limit theorems with a rate of convergence for time-dependent intermittent maps, arXiv E-Prints, arXiv: 1811.11170. [6] M. Nicol, A. Török and S. Vaienti, Central limit theorems for sequential and random intermittent dynamical systems, Ergodic Theory Dynam. Systems, 38 (2018), 1127-1153.  doi: 10.1017/etds.2016.69. [7] D. J. Scott and R. M. Huggins, On the embedding of processes in Brownian motion and the law of the iterated logarithm for reverse martingales, Bull. Austral. Math. Soc., 27 (1983), 443-459.  doi: 10.1017/S0004972700025946. [8] V. G. Sprindžuk, Metric Theory of Diophantine Approximations, Translated from the Russian and edited by Richard A. Silverman, With a foreword by Donald J. Newman, Scripta Series in Mathematics. V. H. Winston & Sons, Washington, D. C., A Halsted Press Book, John Wiley & Sons, New York-Toronto, Ont.-London, 1979.

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##### References:
 [1] R. Aimino, H. Y. Hu, M. Nicol, A. Török and S. Vaienti, Polynomial loss of memory for maps of the interval with a neutral fixed point, Discrete Contin. Dyn. Syst., 35 (2015), 793-806.  doi: 10.3934/dcds.2015.35.793. [2] C. Cuny and F. Merlevède, Strong invariance principles with rate for "reverse" martingale differences and applications, J. Theoret. Probab., 28 (2015), 137-183.  doi: 10.1007/s10959-013-0506-z. [3] D. Dragičević, G. Froyland, C. González-Tokman and S. Vaienti, Almost sure invariance principle for random piecewise expanding maps, Nonlinearity, 31 (2018), 2252-2280.  doi: 10.1088/1361-6544/aaaf4b. [4] N. Haydn, M. Nicol, A. Török and S. Vaienti, Almost sure invariance principle for sequential and non-stationary dynamical systems, Trans. Amer. Math. Soc., 369 (2017), 5293-5316.  doi: 10.1090/tran/6812. [5] O. Hella and J. Leppänen, Central limit theorems with a rate of convergence for time-dependent intermittent maps, arXiv E-Prints, arXiv: 1811.11170. [6] M. Nicol, A. Török and S. Vaienti, Central limit theorems for sequential and random intermittent dynamical systems, Ergodic Theory Dynam. Systems, 38 (2018), 1127-1153.  doi: 10.1017/etds.2016.69. [7] D. J. Scott and R. M. Huggins, On the embedding of processes in Brownian motion and the law of the iterated logarithm for reverse martingales, Bull. Austral. Math. Soc., 27 (1983), 443-459.  doi: 10.1017/S0004972700025946. [8] V. G. Sprindžuk, Metric Theory of Diophantine Approximations, Translated from the Russian and edited by Richard A. Silverman, With a foreword by Donald J. Newman, Scripta Series in Mathematics. V. H. Winston & Sons, Washington, D. C., A Halsted Press Book, John Wiley & Sons, New York-Toronto, Ont.-London, 1979.
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