本文考虑一类$(Tx, g_x(t)) : X \times \mathbb{R} \to X \times \mathbb{R}$形式的斜积，其中$T$是一个连续扩张Markov映射，$g_x : \mathbb{R} \to \mathbb{R}$是$\mathbb{R}$的同胚族.一个函数$u: X \to \mathbb{R}$被称为不变图，如果$\mathrm{graph}(u) = \{(x,u(x)) \mid x\in X\}$是斜积的不变集;等价地，如果$u(T(x)) = g_x(u(x))$。一个研究得很好的问题是考虑这些函数的存在性、正则性和维数理论性质，通常是在纤维方向上的强收缩或膨胀条件下（根据李雅普诺夫指数或部分双曲性）。在这里，我们考虑这样的问题中的设置在纤维方向上的李雅普诺夫指数为零的一组周期性轨道。我们证明，$u$要么有一个“准图”（或“骨感图”）的结构或光滑的动力学，我们给出了一个标准，这种情况发生。
英文摘要：
We consider a family of skew-products of the form $(Tx, g_x(t)) : X \times \mathbb{R} \to X \times \mathbb{R}$ where $T$ is a continuous expanding Markov map and $g_x : \mathbb{R} \to \mathbb{R}$ is a family of homeomorphisms of $\mathbb{R}$. A function $u: X \to \mathbb{R}$ is said to be an invariant graph if $\mathrm{graph}(u) = \{(x,u(x)) \mid x\in X\}$ is an invariant set for the skew-product; equivalently if $u(T(x)) = g_x(u(x))$. A well-studied problem is to consider the existence, regularity and dimension-theoretic properties of such functions, usually under strong contraction or expansion conditions (in terms of Lyapunov exponents or partial hyperbolicity) in the fibre direction. Here we consider such problems in a setting where the Lyapunov exponent in the fibre direction is zero on a set of periodic orbits. We prove that $u$ either has the structure of a `quasi-graph' (or `bony graph') or is as smooth as the dynamics, and we give a criteria for this to happen.
Noise-induced order is the phenomenon by which the chaotic regime of a deterministic system is destroyed in the presence of noise. In this manuscript, we establish noise-induced order for a natural class of systems of dimension $\geq 2$ consisting of a fiber-contracting skew product a over nonuniformly-expanding 1-dimensional system.
We establish a linearization criterion for skew products of contractions in any dimension. We prove their smooth or holomorphic parameter dependence. In the smooth setting, we use the language of tame Fréchet spaces. We apply our result to the linearization of totally projectively expanding Cantor sets.
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Step skew products with interval fibres and a subshift as a base are considered. It is proved that if the fibre maps are continuous, piecewise monotone, expanding and surjective and the subshift has the specification property and a periodic orbit such that the composition of the fibre maps along this orbit is mixing, then the corresponding step skew product has the specification property.
In this paper we show how to apply classical probabilistic tools for partial sums $\sum_{j=0}^{n-1}φ\circτ^j$ generated by a skew product $τ$, built over a sufficiently well mixing base map and a random expanding dynamical system. Under certain regularity assumptions on the observable $φ$, we obtain a central limit theorem (CLT) with rates, a functional CLT, an almost sure invariance principle (ASIP), a moderate deviations principle, several exponential concentration inequalities and Rosenthal type moment estimates for skew products with $α, ϕ$ or $ψ$ mixing base maps and expanding on the aver…
We prove a Berry-Esseen theorem, a local central limit theorem and (local) large and (global) moderate deviations principles for i.i.d. (uniformly) random non-uniformly expanding or hyperbolic maps with exponential first return times. Using existing results the problem is reduced to certain random (Young) tower extensions, which is the main focus of this paper. On the random towers we will obtain our results using contraction properties of random complex equivariant cones with respect to the complex Hilbert projective metric.
