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

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