任意S-单峰映射的大偏差率函数的存在性

周期性无法保证旋转存在的存在性：在$\mathbb{T}^3$上的反例

When Periodicity Fails to Guarantee the Existence of Rotation: A Counterexample on the 3-torus with a Nilpotent Linearization

本文构造了一个光滑周期动力系统反例，证明在弱意义下旋转向量存在，但在强意义下不存在，挑战了光滑性和周期性对有界偏差的保证。

双曲曲面上同调增长率的偏差大数定律

Large deviations of homological growth rates for hyperbolic surfaces

本文研究双曲曲面上有向测地线同调增长率的大偏差行为，证明了速率函数的存在性，并建立了 Erdős-Rényi 定律和改进的大偏差上界。

无吉布斯态的可数马尔可夫移位的2级大偏差原理

Level-2 large deviation principle for countable Markov shifts without Gibbs states

本文在无Bowen吉布斯态假设下，通过诱导充分条件保证周期构型Borel概率测度列的指数紧性，建立了单侧可数全移位的2级大偏差原理，并将其应用于Rényi映射连分数展开，得到对应大偏差原理及二次无理数的加权等分布结果。

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…

Large deviations for stochastic nonlinear systems of slow-fast diffusions with non-Gaussian Lévy noises

We establish the large deviation principle for the slow variables in slow-fast dynamical system driven by both Brownian noises and Lévy noises. The fast variables evolve at much faster time scale than the slow variables, but they are fully inter-dependent. We study the asymptotics of the logarithmic functionals of the slow variables in the three regimes based on viscosity solutions to the Cauchy problem for a sequence of partial integro-differential equations. We also verify the comparison principle for the related Cauchy problem to show the existence and uniqueness of the limit for viscosity …

From Rates of mixing to recurrence times via large deviations

A classic approach in dynamical systems is to use particular geometric structures to deduce statistical properties, for example the existence of invariant measures with stochastic-like behaviour such as large deviations or decay of correlations. Such geometric structures are generally highly non-trivial and thus a natural question is the extent to which this approach can be applied. In this paper we show that in many cases stochastic-like behaviour itself implies that the system has certain non-trivial geometric properties, which are therefore necessary as well as sufficient conditions for the…

单峰可测伪阿诺索夫映射

Unimodal Measurable Pseudo-Anosov Maps

本文构造球面上连续变化的同胚族，每个映射为可测伪阿诺索夫映射，半共轭于帐篷映射，具有拓扑传递性、遍历性、稠密周期点及熵 log λ，且典型参数下湍流处处非正则。

平稳密度与拟平稳密度的拟势：存在性、正则性及应用

Quasi-potential of stationary and quasi-stationary densities: existence, regularity, and applications

本文研究随机扰动动力系统中平稳与拟平稳分布密度的拟势，在最大吸引子吸引域内的正不变集框架下，探讨其存在性与正则性，并指出广泛应用前景。

Large Deviation Principle for arithmetic functions in continued fraction expansion

Khinchin proved that the arithmetic mean of continued fraction digits of Lebesgue almost every irrational number in $(0,1)$ diverges to infinity. Hence, none of the classical limit theorems such as the weak and strong laws of large numbers or central limit theorems hold. Nevertheless, we prove the existence of a large deviations rate function which estimates exponential probabilities with which the arithmetic mean of digits stays away from infinity.

Large deviations for equilibrium measures and selection of subaction

Given a Lipschitz function $f:\{1,...,d\}^\mathbb{N} \to \mathbb{R}$, for each $β>0$ we denote by $μ_β$ the equilibrium measure of $βf$ and by $h_β$ the main eigenfunction of the Ruelle Operator $L_{βf}$. Assuming that $\{μ_β\}_{β>0}$ satisfy a large deviation principle, we prove the existence of the uniform limit $V= \lim_{β\to\infty}\frac{1}β\log(h_β)$. Furthermore, the expression of the deviation function is determined by its values at the points of the union of the supports of maximizing measures. We study a class of potentials having two ergodic maximizing measures and prove that a L.D.P.…

Deviations of Ergodic sums for Toral Translations I : Convex bodies

We show the existence of a limiting distribution $\cD_\cC$ of the adequately normalized discrepancy function of a random translation on a torus relative to a strictly convex set $\cC$. Using a correspondence between the small divisors in the Fourier series of the discrepancy function and lattices with short vectors, and mixing of diagonal flows on the space of lattices, we identify $\cD_\cC$ with the distribution of the level sets of a function defined on the product of the space of lattices with an infinite dimensional torus. We apply our results to counting lattice points in slanted cylinder…