The size of the chain recurrent set for generic maps on an $n$-dimensional locally $(n-1)$-connected compact space

Tom 119 / 2010

Katsuya Yokoi Colloquium Mathematicum 119 (2010), 229-236 MSC: Primary 37B20; Secondary 37C20, 37A05. DOI: 10.4064/cm119-2-5


For $n \geq 1$, given an $n$-dimensional locally $(n-1)$-connected compact space $X$ and a finite Borel measure $\mu$ without atoms at isolated points, we prove that for a generic (in the uniform metric) continuous map $f:X \to X$, the set of points which are chain recurrent under $f$ has $\mu$-measure zero. The same is true for $n =0$ (skipping the local connectedness assumption).


  • Katsuya YokoiDepartment of Mathematics
    Shimane University at Matsue
    Matsue, 690-8504, Japan
    Department of Mathematics
    Jikei University School of Medicine
    Chofu, Tokyo 182-8570, Japan

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