Recurrent and periodic points in dendritic Julia sets (Q2845434)

Let \(X\) be a compact metric space and let \(g:X\to X\) be continuous. The closure of the set of all recurrent points of \(g\) is called the center of \(g\). A theorem of Sharkovskiy says that if \(X\) is an interval, then the center of \(g\) coincides with the closure of the set of periodic points of \(g\). Here a result of this type is obtained for the case when \(X\) is a dendrite. (A dendrite is a non-degenerate locally connected continuum that does not contain Jordan curves.) It is shown that all recurrent points with a certain property (``arc type'') are contained in the closure of the set of periodic points. It is noted that if \(X\) is a finite tree, then all recurrent points are of arc type, so the result is a generalisation of Sharkovskiy's theorem. The results are applied to polynomials with dendritic Julia sets.