AI 中文总结
研究S-单峰映射李雅普诺夫指数，证明非平坦临界点的该映射对几乎每个勒贝格点李雅普诺夫指数存在且为常数λ_T，给出其为0的充要条件，还得出相关特殊映射的结论。
AI 中文摘要
本文表明，对于[0,1]上任何具有非平坦临界点的S-单峰映射T，勒贝格几乎每个点的李雅普诺夫指数存在且等于常数λ_T∈R。此外，λ_T = 0当且仅当T既没有具有正熵的绝对连续T-不变概率测度，也没有严格稳定的周期轨道。
英文摘要
In this paper, we show that for any $S$-unimodal map $T$ on $[0,1]$ with a non-flat critical point the Lyapunov exponent exists for Lebesgue almost every point and is equal to a constant $λ_T\in\mathbb{R}$. Moreover, $λ_T=0$ if and only if $T$ admits neither an absolutely continuous $T$-invariant probability measure with positive entropy nor a strictly stable periodic orbit. Consequently, if an $S$-unimodal map with a non-flat critical point is infinitely renormalizable or non-statistical then for Lebesgue almost every $x\in [0,1]$ the Lyapunov exponent along the orbit of $x$ exists and is equal to $0$. A key ingredient is the following result of independent interest. If an $S$-unimodal map with a non-flat critical point has no periodic attractor then for Lebesgue almost every $x\in [0,1]$ the lower Lyapunov exponent along the orbit of $x$ is non-negative. This shows that, in the absence of periodic attractors, exponential contraction cannot occur along the orbit of Lebesgue almost every point.
Comments26 pages
