A Spectral Approach for Quenched Limit Theorems for Random Expanding Dynamical Systems

• Published in: Communications in mathematical physics Vol. 360; no. 3; pp. 1121 - 1187
• Main Authors: Dragičević, D., Froyland, G., González-Tokman, C., Vaienti, S.
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01-06-2018; Springer Nature B.V; Springer Verlag
• Subjects: Classical and Quantum Gravitation; Complex Systems; Dynamical Systems; Ergodic processes; Liapunov exponents; Mathematical and Computational Physics; Mathematical Physics; Mathematics; Physics; Physics and Astronomy; Quantum Physics; Quenching; Random variables; Relativity Theory; Spectral methods; Theorems; Theoretical
• ISSN: 0010-3616; 1432-0916
• DOI: 10.1007/s00220-017-3083-7

We prove quenched versions of (i) a large deviations principle (LDP), (ii) a central limit theorem (CLT), and (iii) a local central limit theorem for non-autonomous dynamical systems. A key advance is the extension of the spectral method, commonly used in limit laws for deterministic maps, to the general random setting. We achieve this via multiplicative ergodic theory and the development of a general framework to control the regularity of Lyapunov exponents of
twisted transfer operator cocycles
with respect to a twist parameter. While some versions of the LDP and CLT have previously been proved with other techniques, the local central limit theorem is, to our knowledge, a completely new result, and one that demonstrates the strength of our method. Applications include non-autonomous (piecewise) expanding maps, defined by random compositions of the form
T
σ
n
-
1
ω
∘
⋯
∘
T
σ
ω
∘
T
ω
. An important aspect of our results is that we only assume ergodicity and invertibility of the random driving
σ
:
Ω
→
Ω
; in particular no expansivity or mixing properties are required.