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 Markovmap 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 graphif $\mathrm{graph}(u) = \{(x,u(x)) \mid x\in X\}$ is an invariant set for theskew-product; equivalently if $u(T(x)) = g_x(u(x))$. A well-studied problem isto consider the existence, regularity and dimension-theoretic properties ofsuch functions, usually under strong contraction or expansion conditions (interms of Lyapunov exponents or partial hyperbolicity) in the fibre direction.Here we consider such problems in a setting where the Lyapunov exponent in thefibre direction is zero on a set of periodic orbits. We prove that $u$ eitherhas the structure of a `quasi-graph' (or `bony graph') or is as smooth as thedynamics, and we give a criteria for this to happen.
展开▼
机译：我们考虑一个形式为$（Tx，g_x（t））的偏乘积族：X \ times \ mathbb {R} \ to X \ times \ mathbb {R} $其中$ T $是连续扩展的Markovmap和$ g_x：\ mathbb {R} \ to \ mathbb {R} $是$ \ mathbb {R} $的同胚族。如果$ \ mathrm {graph}（u）= \ {（x，u（x））\ mid x \ in X \} $，则函数$ u：X \ to \ mathbb {R} $被称为不变图。是偏积的不变集；等效地，如果$ u（T（x））= g_x（u（x））$。一个经过充分研究的问题是要考虑此类函数的存在，规律性和尺寸理论性质，通常是在纤维方向上的强收缩或扩张条件下（李雅普诺夫指数的中间值或部分双曲率）。在一组周期性轨道上，纤维方向的李雅普诺夫指数为零。我们证明$ u $具有“准图”（或“ bony graph”）的结构，或者与动力学一样平滑，并且我们为此提供了条件。
展开▼
