In the current paper we suggest a new robust algorithm to search for cycles of arbitrary length in non-linear autonomous discrete dynamical systems. With the help of the computer we were able to find (unstable) cycles for several basic maps of nonlinear science: Henon, Holmes cubic, Ikeda, Lozi, Elhaj-Sprott. The theoretical part of the paper is based on properties of a new family of extremal polynomials that contains the Fejer and Suffridge polynomials. The associated combination of geometric complex analysis and discrete dynamics seems to be a new phenomenon, both theoretical and practical. A novelty of this paper is in the discovery of a close connection between two seemingly disconnected fields: extremal polynomials and cycles in dynamical systems.

We are thankful to an anonymous referee for carefully reading the paper and finding several typos. M.T. was supported in part by the NSF grant DMS--1636435.