σ有限不变测度的存在性及其在实一维动力学中的应用
The existence of sigma-finite invariant measures, applications to real one-dimensional dynamics
摘要：
给出了$σ-$有限绝对连续不变测度的一般构造.我们将证明，$f^n_*(λ)$的Radon-Nykodym导数的局部有界畸变将意味着对于关于$λ$绝对连续的映射$f$存在$σ-$有限不变测度，它是描述测度零集合的相空间上的测度。进一步讨论了实一维动力系统的$σ-$有限不变绝对连续测度存在的充分条件。
英文摘要：
A general construction for $σ-$finite absolutely continuous invariant measure will be presented. It will be shown that the local bounded distortion of the Radon-Nykodym derivatives of $f^n_*(λ)$ will imply the existence of a $σ-$finite invariant measure for the map $f$ which is absolutely continuous with respect to $λ$, a measure on the phase space describing the sets of measure zero. Furthermore we will discuss sufficient conditions for the existence of $σ-$finite invariant absolutely continuous measures for real 1-dimensional dynamical systems.
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