关于$1$维映射不变测度的一些结果

Pub Date : 2021-09-01 DOI:10.3836/tjm/1502179353
F. Schweiger
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引用次数: 0

摘要

对于许多纤维系统,可以证明不变测度的存在,但对密度的形状知之甚少。本文讨论了不变密度的各种例子:具有四个分支的分段分数线性映射和与具有递增数字的连续分数相关的映射。存在具有不可积密度的遍历映射,其不具有无关的不动点,并且映射使得错过数字$k=1$的点集具有正Lebesgue测度。
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Some Results on Invariant Measures for $1$-dimensional Maps
For many fibred systems the existence of an invariant measure can be proved but considerably less is known about the shape of the density. In this note various examples of invariant densities are discussed: Piecewise fractional linear maps with four branches and maps which are associated to continued fractions with increasing digits. There are ergodic maps with a non-integrable density which do not have an indifferent fixed point and maps such that the set of points which miss the digit $k=1$ has positive Lebesgue measure.
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