Dynamical Systems Seminar

For a continuous function f from an interval I to itself let t_n(m) be the minimal number of periodic points of f of minimal order m, given it has a periodic point of minimal order n. Sharkovsky's theorem states that t_n(m)\ge 1 if and only if m follows n in the Sharkovsky's ordering. In the talk we give an explicit description of the sequence t_n(m). Based on work with J. Byszewski.

We study the dynamics of continuous maps on compact metric spaces containing a free interval (an open subset homeomorphic to the interval (0,1)). We provide a new proof of a result of Dirbák, Snoha and Špitalský stating that every continuous and transitive, non-minimal map of a space with a free interval is relatively mixing, non-invertible, has positive topological entropy, and dense periodic points. Joint work with D. Kwietniak.