Recent results on iterative roots

Abstract

Abstract. This is a rough survey of some results on iterative roots (fractional iterates) published recently. Also some historical information to clear the connection to previous results has been given.The main topics are: the conjugacy of piecewise monotonic functions and their iterative roots, stability of iterative roots, some substitutes and generalizations of the notion of iterative root using set-valued functions.Résumé. Le présent article de revue approximative contient de certains résultats sur les racines itératives (itérées fractionnaires) publiés récemment. Une information historique est aussi donnée pour montrer la liason avec des résultats précédents. Les sujets principaux sont: la conjugaison des fonctions monotones par morceaux et leurs racines itératives, stabilité des racines itératives, certains substituts et généralizations de la notion de la racine itérative qui utilisent les fonctions multivaluées.

The non-iterates are dense in the space of continuous self-maps

“…For n = 2, a solution of (1.1) is called an iterative square root of f. The iterative root problem (1.1), rooted in the classic work [2] of Babbage, is of great interest because it is a weak version of the invariant curve problem [22] and the embedding flow problem [12] of dynamical systems, and has numerous applications in, for example, informatics [20] and neural networks [16]. Many of the findings on this problem in various aspects are included in the monographs [21,22], the book [35], and the survey papers [3,18,36,39], with some of the most recent results also available in [4, 11, 23-25, 27-29, 37, 38]. It turns out that even very simple and 'dynamically rich' functions can have no iterative roots.…”