Quasistatic Dynamics with Intermittency

Abstract

We study an intermittent quasistatic dynamical system composed of nonuniformly hyperbolic Pomeau-Manneville maps with time-dependent parameters. We prove an ergodic theorem which shows almost sure convergence of time averages in a certain parameter range, and identify the unique physical family of measures. The theorem also shows convergence in probability in a larger parameter range. In the process, we establish other results that will be useful for further analysis of the statistical properties of the model.We use the usual floor and ceiling functions for rounding real numbers x to the nearest integer: ⌊x⌋ = max{n ∈ N : n ≤ x} and ⌈x⌉ = min{n ∈ N : n ≥ x}.System constants, whose values are determined solely by the parameters defining the model, are distinguished by a subindex (e.g., C 0 ) from generic constants (e.g., C), whose values may change from one expression to the next.

Central limit theorems with a rate of convergence for time-dependent intermittent maps

“…given that the limit exists. The previous papers [13,21,30,31,43] dealt with statistical properties of QDSs. In [21], the CLT was established for a class of QDSs constructed over uniformly expanding circle maps.…”

“…Recall that a QDS is a pair (T, γ) where T = {T n,k : 0 ≤ k ≤ n, n ∈ N} is a triangular array of maps in a topological space M, and γ : [0, 1] → M is a curve such that T n,⌊nt⌋ → γ t as n → ∞. In [31], the following intermittent version of the QDS was introduced. For clarity we recast some of the definitions introduced in Section 2.3 for the intermittent QDS.…”

“…By Lemma 2.4 in [29], for each l with m + 1 ≤ l ≤ m + k, there are functions g 1 , g 2 ∈ C * (β * ), such that L τ (l−1)/n · · · L τ (m+1)/n gf p α = g 1 − g 2 , and g i 1 ≤ C(β * )( f Lip +1) for some constant C(β * ) > 0 depending only on T β * . Theorem 5.1 in [31] applies to cone functions, and it follows that…”

Extreme Value Laws for sequences of intermittent maps

“…The latter was successively generalized in the quenched case (with respect to the stationary measure and for almost all the realizations), by Nicol, Torok and Vaienti [NTV16]; this paper contains also a proof of the central limit theorem for sequential systems and its results will be used again in the next section. Still in this context we also quote the paper by Leppänen and Stenlund [LS15] where a few results on the continuity of the densities and their pushforward with respect to the parameter α are proved.…”

Almost Sure Rates of Mixing for Random Intermittent Maps

“…Dedicated to Abdulla Azamov and Leonid Bunimovich on the occasion of their 70th birthday 1 A subclass of the so called Pomeau-Manneville maps introduced in [18], and popularised by Liverani, Saussol and Vaienti in [15]. Such systems have attracted attention of both mathematicians and physicists (see [14] for a recent work in this area).…”