Decay of Correlations for Unbounded Observables

A note on large deviations for unbounded observables

We consider exponential large deviations estimates for unbounded observables on uniformly expanding dynamical systems. We show that uniform expansion does not imply the existence of a rate function for unbounded observables no matter the tail behavior of the cumulative distribution function. We give examples of unbounded observables with exponential decay of autocorrelations, exponential decay under the transfer operator in each $L^p$, $1\le p < \infty$, and strictly stretched exponential large deviation. For observables of form $|\log d(x,p)|^α$, $p$ periodic, on uniformly expanding systems w…

动力系统中某些无界观测量的大偏差估计

Large deviations for some unbounded observables in dynamical systems

本文研究了动力系统中强混饨马尔可夫链的大偏差估计，并推导了局部常数随机协-cycle和扩展映射的无界观测量的大偏差估计。

On the convergence to equilibrium of unbounded observables under a family of intermittent interval maps

We consider a family $\{ T_{r} \colon [0, 1] \circlearrowleft \}_{r \in [0, 1]}$ of Markov interval maps interpolating between the Tent map $T_{0}$ and the Farey map $T_{1}$. Letting $\mathcal{P}_{r}$ denote the Perron-Frobenius operator of $T_{r}$, we show, for $β\in [0, 1]$ and $α\in (0, 1)$, that the asymptotic behaviour of the iterates of $\mathcal{P}_{r}$ applied to observables with a singularity at $β$ of order $α$ is dependent on the structure of the $ω$-limit set of $β$ with respect to $T_{r}$. Having a singularity it seems that such observables do not fall into any of the function cla…

关于一类无界观测量的极限定理及其在"采样林德勒夫假设"中的应用

Limit Theorems for a class of unbounded observables with an application to "Sampling the Lindelöf hypothesis"

本文证明了针对一类无界强振荡观测量的极限定理，并应用于黎曼zeta函数的采样林德勒夫假设研究。

First hyperbolic times for intermittent maps with unbounded derivative

We establish some statistical properties of the hyperbolic times for a class of nonuniformly expanding dynamical systems. The maps arise as factors of area preserving maps of the unit square via a geometric Baker's map type construction, exhibit intermittent dynamics, and have unbounded derivatives. The geometric approach captures various examples from the literature over the last thirty years. The statistics of these maps are controlled by the order of tangency that a certain "cut function" makes with the boundary of the square. Using a large deviations result of Melbourne and Nicol we obtain…

Non-exponential stability and decay rates in nonlinear stochastic difference equation with unbounded noises

We consider stochastic difference equation x_{n+1} = x_n (1 - h f(x_n) + \sqrt{h} g(x_n) ξ_{n+1}), where functions f and g are nonlinear and bounded, random variables ξ_i are independent and h>0 is a nonrandom parameter. We establish results on asymptotic stability and instability of the trivial solution x_n=0. We also show, that for some natural choices of the nonlinearities f and g, the rate of decay of x_n is approximately polynomial: we find α>0 such that x_n decay faster than n^{-α+ε} but slower than n^{-α-ε} for any ε>0. It also turns out that if g(x) decays faster than f(x) as x->0, the…