Henk Bruin, Mark Holland, Matthew Nicol
We prove measurable Livsic theorems for dynamical systems modelled by Markov Towers. Our regularity results apply to solutions of cohomological equations posed on Henon-like mappings and a wide variety of nonuniformly hyperbolic systems. We consider both Holder cocycles and cocycles with singularities of prescribed order.
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General real measurable Livšic regularity via transfer operators
该研究借助转移算子的抽象假设，证明了非可逆动力系统上实值上循环的一般可测Livšic正则性定理，并将其应用于三类动力系统得到对应正则性结果。
We prove a nonuniformly hyperbolic version Livšic theorem, with cocycles taking values in the group of invertible bounded linear operators on a Banach space. The result holds without the ergodicity assumption of the hyperbolic measure. Moreover, We also prove a $μ$-continuous solution of the cohomological equation is actually Hölder continuous for the uniform hyperbolic system.
We consider the regularity of measurable solutions $χ$ to the cohomological equation \[ ϕ= χ\circ T -χ, \] where $(T,X,μ)$ is a dynamical system and $ϕ\colon X\rightarrow \R$ is a $C^k$ valued cocycle in the setting in which $T \colon X\rightarrow X$ is a piecewise $C^k$ Gibbs--Markov map, an affine $β$-transformation of the unit interval or more generally a piecewise $C^{k}$ uniformly expanding map of an interval. We show that under mild assumptions, bounded solutions $χ$ possess $C^k$ versions. In particular we show that if $(T,X,μ)$ is a $β$-transformation then $χ$ has a $C^k$ version, thus…
A Livšic-type theorem and some regularity properties for nonadditive sequences of potentials
本文研究渐近可加序列的上同调，证明势的几乎可加序列的Livšic型定理，据此刻画几乎可加序列、构造非物理等价于Hölder连续可加序列的例子，并提出几乎可加序列的分类方案。
We prove a Livsic type theorem for cocycles taking values in groups of diffeomorphisms of low-dimensional manifolds. The results hold without any localization assumption and in very low regularity. We also obtain a general result (in any dimension) which gives necessary and sufficient conditions to be a coboundary.
We prove that solutions $u$ of the equation $f=u-u\circ T$ are automatically Holder continuous when $f$ is Holder continuous, and $T$ is non uniformly expanding and Markov. This result applies in particular to Young towers and to intermittent maps.
We consider processes which are functions of finite-state Markov chains. It is well known that such processes are rarely Markov. However, such processes are often regular in the following sense: the distant past values of the process have diminishing influence on the distribution of the present value. In the present paper, we present novel sufficient conditions for regularity of functions of Markov chains.
Decay of Correlations via induced Weak Gibbs Markov maps for non-Hölder observables
本文针对具有诱导弱吉布斯-马尔可夫映射的动力系统，在比Hölder更弱的连续模条件下，建立了更广泛观测量的相关性衰减、中心极限定理和大偏差估计。
We investigate limit theorems for Birkhoff sums of locally Hölder functions under the iteration of Gibbs-Markov maps. Aaronson and Denker have given sufficient conditions to have limit theorems in this setting. We show that these conditions are also necessary: there is no exotic limit theorem for Gibbs-Markov maps. Our proofs, valid under very weak regularity assumptions, involve weak perturbation theory and interpolation spaces. For L^2 observables, we also obtain necessary and sufficient conditions to control the speed of convergence in the central limit theorem.
Statistical properties of Markov shifts (part I)
本文证明了非齐次马尔可夫链部分和的中心极限定理、Berry-Esseen型定理、大偏差等统计性质，并指出该结果对平稳马尔可夫链和非平稳伯努利平移同样具有创新性。
Representation of Markov chains by random maps: existence and regularity conditions
本文利用最优传输理论系统研究用随机映射族表示马尔可夫链的问题，通过 Legendre 变换建立测度支撑集凸性与映射正则性的联系，并给出可测、连续及微分同胚表示的条件。
In this paper we prove a multifractal formalism of Birkhoff averages for interval maps with countably many branches. Furthermore, we prove that under certain regularity assumptions on the potential the Birkhoff spectrum is real analytic. Applications of these results to number theory are also given. Finally, we compute the Hausdorff dimension of the set of points for which the Birkhoff average is infinite.
Ergodicity and regularity properties of ODEs with semi-Markov switching
该文研究有限向量场间按任意 R_+ 分布随机切换形成的 ODE 过程，证明非奇异分布下过程具有 Feller 性，并给出使 Markov 情形的 Lie 括号条件仍蕴含遍历性的条件。
In skew-product systems with contractive factors, all orbits asymptotically approach the graph of the so-called sync function; hence, the corresponding regularity properties primarily matter. In the literature, sync function Lipschitz continuity and differentiability have been proved to hold depending on the derivative of the base reciprocal, if not on its Lyapunov exponent. However, forcing topological features can also impact the sync function regularity. Here, we estimate the total variation of sync functions generated by one-dimensional Markov maps. A sharp condition for bounded variation …
We prove that, up to topological conjugacy, every Smale space admits an Ahlfors regular Bowen measure. Bowen's construction of Markov partitions implies that Smale spaces are factors of topological Markov chains. The latter are equipped with Parry's measure which is Ahlfors regular. By extending Bowen's construction we create a tool for transferring the Ahlfors regularity of the Parry measure down to the Bowen measure of the Smale space. An essential part of our method uses a refined notion of approximation graphs over compact metric spaces. Moreover, we obtain new estimates for the Hausdorff,…
Markovian reduction and exponential mixing in total variation for random dynamical systems
本文研究了有限维紧致相空间下由随机噪声驱动的离散时间动力系统的大时间行为，通过马尔可夫约化和Markov过程混合理论，证明了轨迹在全变差度量下收敛于唯一测度，并将该结果推广至平稳噪声驱动的系统。
We put forward a new method for obtaining quantitative lower bounds on the top Lyapunov exponent of stochastic differential equations (SDEs). Our method combines (i) an (apparently new) identity connecting the top Lyapunov exponent to a Fisher information-like functional of the stationary density of the Markov process tracking tangent directions with (ii) a novel, quantitative version of Hörmander's hypoelliptic regularity theory in an $L^1$ framework which estimates this (degenerate) Fisher information from below by a $W^{s,1}_{\mathrm{loc}}$ Sobolev norm. This method is applicable to a wide …
Statistical Limit Theorems for Axiom A Diffeomorphisms: Exponential Mixing, Central Limit Theorem, and Large Deviations
本文研究Axiom A微分同胚的平衡态的统计极限定理，通过Part I中建立的Ruelle转移算子的谱间隙和Part III中的马尔可夫分区编码，证明了体积引理、指数相关衰减、中心极限定理、几乎必然不变原理以及大偏差原理，并展示了这些结果如何从单一的谱机制中推导出来，明确依赖于双曲性数据。
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