Random Composition of L-S-V Maps Sampled Over Large Parameter Ranges
| Autor: | Christopher Bose, Matteo Tanzi, Anthony Quas |
|---|---|
| Jazyk: | angličtina |
| Rok vydání: | 2020 |
| Předmět: |
Limit of a function
Applied Mathematics 010102 general mathematics Probability (math.PR) General Physics and Astronomy Statistical and Nonlinear Physics Dynamical Systems (math.DS) Composition (combinatorics) Absolute continuity Fixed point 01 natural sciences Omega 010101 applied mathematics Combinatorics Sampling distribution 37H05 (primary) 37D25 37A50 (secondary) Piecewise FOS: Mathematics Differentiable function 0101 mathematics Mathematics - Dynamical Systems Mathematical Physics Mathematics - Probability Mathematics |
| Popis: | Liverani–Saussol–Vaienti (L–S–V) maps form a family of piecewise differentiable dynamical systems on [0, 1] depending on one parameter ω ∈ R + . These maps are everywhere expanding apart from a neutral fixed point. It is well known that depending on the amount of expansion close to the neutral point, they have either an absolutely continuous invariant probability measure and polynomial decay of correlations (ω < 1), or a unique physical measure that is singular and concentrated at the neutral point (ω > 1). In this paper, we study the composition of L–S–V maps whose parameters are randomly sampled from a range in R + , and where these two contrasting behaviours are mixed. We show that if the parameters ω < 1 are sampled with positive probability, then the stationary measure of the random system is absolutely continuous; the annealed decay rate of correlations is close (or in some cases equal) to the fastest rate of decay among those of the sampled systems; and suitably rescaled Birkhoff averages converge to limit laws. In contrast to previous studies where ω ∈ [0, 1], we allow ω > 1 in our sampling distribution. We also show that one can obtain similar decay of correlation rates for ω ∈ R + , when sampling is done with respect to a family of smooth, heavy-tailed distributions. |
| Databáze: | OpenAIRE |
| Externí odkaz: |
|