scholarly journals Rates in almost sure invariance principle for quickly mixing dynamical systems

2019 ◽  
Vol 20 (01) ◽  
pp. 2050002
Author(s):  
C. Cuny ◽  
J. Dedecker ◽  
A. Korepanov ◽  
F. Merlevède
Keyword(s):  
Brownian Motion ◽  
Dynamical Systems ◽  
Large Class ◽  
Exponential Decay ◽  
Invariance Principle ◽  
Decay Of Correlations ◽  
Hölder Continuous ◽  
One Dimensional ◽  
Exponential Decay Of Correlations ◽  
Stretched Exponential Decay

For a large class of quickly mixing dynamical systems, we prove that the error in the almost sure approximation with a Brownian motion is of order [Formula: see text] with [Formula: see text]. Specifically, we consider nonuniformly expanding maps with exponential and stretched exponential decay of correlations, with one-dimensional Hölder continuous observables.

scholarly journals Rates in almost sure invariance principle for slowly mixing dynamical systems

2019 ◽  
Vol 40 (9) ◽  
pp. 2317-2348 ◽  
Author(s):  
C. CUNY ◽  
J. DEDECKER ◽  
A. KOREPANOV ◽  
F. MERLEVÈDE
Keyword(s):  
Markov Chain ◽  
Dynamical Systems ◽  
Invariance Principle ◽  
Slowly Varying Function ◽  
Hölder Continuous ◽  
One Dimensional ◽  
Deterministic Dynamical Systems ◽  
Almost Sure Invariance Principle ◽  
The One ◽  
Dependent Processes

We prove the one-dimensional almost sure invariance principle with essentially optimal rates for slowly (polynomially) mixing deterministic dynamical systems, such as Pomeau–Manneville intermittent maps, with Hölder continuous observables. Our rates have form $o(n^{\unicode[STIX]{x1D6FE}}L(n))$, where $L(n)$ is a slowly varying function and $\unicode[STIX]{x1D6FE}$ is determined by the speed of mixing. We strongly improve previous results where the best available rates did not exceed $O(n^{1/4})$. To break the $O(n^{1/4})$ barrier, we represent the dynamics as a Young-tower-like Markov chain and adapt the methods of Berkes–Liu–Wu and Cuny–Dedecker–Merlevède on the Komlós–Major–Tusnády approximation for dependent processes.

Statistics of closest return for some non-uniformly hyperbolic systems

2001 ◽  
Vol 21 (2) ◽  
pp. 401-420 ◽  
Author(s):  
P. COLLET
Keyword(s):  
Dynamical Systems ◽  
Exponential Decay ◽  
Reference Point ◽  
Hyperbolic Systems ◽  
Initial Point ◽  
Decay Of Correlations ◽  
Exponential Decay Of Correlations ◽  
Hyperbolic Dynamical Systems ◽  
Uniformly Hyperbolic Systems ◽  
The Law

For non-uniformly hyperbolic maps of the interval with exponential decay of correlations we prove that the law of closest return to a given point when suitably normalized is almost surely asymptotically exponential. A similar result holds when the reference point is the initial point of the trajectory. We use the framework for non-uniformly hyperbolic dynamical systems developed by L. S. Young.

scholarly journals Almost sure mixing rates for non-uniformly expanding maps

2018 ◽  
Vol 18 (04) ◽  
pp. 1850027
Author(s):  
Xin Li ◽  
Helder Vilarinho
Keyword(s):  
Exponential Decay ◽  
Decay Of Correlations ◽  
Expanding Maps ◽  
Critical Set ◽  
Stretched Exponential ◽  
Exponential Decay Of Correlations ◽  
Mixing Rates ◽  
Almost All ◽  
Set Of Points ◽  
Stretched Exponential Decay

We consider random perturbations of non-uniformly expanding maps, possibly having a non-degenerate critical set. We prove that if the volume measure of the set of points failing the non-uniform expansion or the slow recurrence to the critical set, at a certain times has a (stretched) exponential decay for almost all random orbits, then the decay of correlations along random orbits is stretched exponential, up to some waiting time. As applications we obtain almost sure stretched exponential decay of correlations along random orbits for Viana maps, as for a class of non-uniformly expanding local diffeomorphisms and a family of unimodal maps.

scholarly journals Statistical Properties of Lorenz-like Flows, Recent Developments and Perspectives

2014 ◽  
Vol 24 (10) ◽  
pp. 1430028 ◽  
Author(s):  
Vitor Araujo ◽  
Stefano Galatolo ◽  
Maria José Pacifico
Keyword(s):  
Exponential Decay ◽  
Chaotic Behavior ◽  
Decay Of Correlations ◽  
Statistical Properties ◽  
Computer Assisted ◽  
Lorenz Attractor ◽  
Lorenz Equations ◽  
Physical Measure ◽  
Exponential Decay Of Correlations ◽  
Statistical Behavior

We comment on the mathematical results about the statistical behavior of Lorenz equations and its attractor, and more generally on the class of singular hyperbolic systems. The mathematical theory of such kind of systems turned out to be surprisingly difficult. It is remarkable that a rigorous proof of the existence of the Lorenz attractor was presented only around the year 2000 with a computer-assisted proof together with an extension of the hyperbolic theory developed to encompass attractors robustly containing equilibria. We present some of the main results on the statistical behavior of such systems. We show that for attractors of three-dimensional flows, robust chaotic behavior is equivalent to the existence of certain hyperbolic structures, known as singular-hyperbolicity. These structures, in turn, are associated with the existence of physical measures: in low dimensions, robust chaotic behavior for flows ensures the existence of a physical measure. We then give more details on recent results on the dynamics of singular-hyperbolic (Lorenz-like) attractors: (1) there exists an invariant foliation whose leaves are forward contracted by the flow (and further properties which are useful to understand the statistical properties of the dynamics); (2) there exists a positive Lyapunov exponent at every orbit; (3) there is a unique physical measure whose support is the whole attractor and which is the equilibrium state with respect to the center-unstable Jacobian; (4) this measure is exact dimensional; (5) the induced measure on a suitable family of cross-sections has exponential decay of correlations for Lipschitz observables with respect to a suitable Poincaré return time map; (6) the hitting time associated to Lorenz-like attractors satisfy a logarithm law; (7) the geometric Lorenz flow satisfies the Almost Sure Invariance Principle (ASIP) and the Central Limit Theorem (CLT); (8) the rate of decay of large deviations for the volume measure on the ergodic basin of a geometric Lorenz attractor is exponential; (9) a class of geometric Lorenz flows exhibits robust exponential decay of correlations; (10) all geometric Lorenz flows are rapidly mixing and their time-1 map satisfies both ASIP and CLT.

scholarly journals Limiting stochastic processes of shift-periodic dynamical systems

2019 ◽  
Vol 6 (11) ◽  
pp. 191423
Author(s):  
Julia Stadlmann ◽  
Radek Erban
Keyword(s):  
Brownian Motion ◽  
Random Walk ◽  
Dynamical Systems ◽  
Stochastic Processes ◽  
Discrete Time ◽  
Real Line ◽  
Continuous Time ◽  
Initial Distribution ◽  
Dynamical Behaviour ◽  
One Dimensional

A shift-periodic map is a one-dimensional map from the real line to itself which is periodic up to a linear translation and allowed to have singularities. It is shown that iterative sequences x n +1 = F ( x n ) generated by such maps display rich dynamical behaviour. The integer parts ⌊ x n ⌋ give a discrete-time random walk for a suitable initial distribution of x 0 and converge in certain limits to Brownian motion or more general Lévy processes. Furthermore, for certain shift-periodic maps with small holes on [0,1], convergence of trajectories to a continuous-time random walk is shown in a limit.

Sign in / Sign up

Export Citation Format

Share Document