The Borel structure of iterates of continuous functions

Abstract. Let C denote the space of continuous functions mapping [0,1] into itself and endowed with the sup metric. It has been shown that C 2 = {f°f:fe C} is an analytic but non-Borel subset of C. This implies that there is no simple geometric characterization for a function being a square. In this paper we consider the problem of characterizing those functions which can be approximated by squares. In the first section we prove that any continuous function mapping a closed proper subset of [0,1] into [0,1] can be extended to a square. In particular this shows that C 2 is L p dense in C. On the other hand, C 2 does not contain a ball when C is endowed with the sup metric. In the second section we prove that no strictly decreasing function can be uniformly approximated by squares, although the distance between the class of strictly decreasing functions and C 2 is zero. In the last section we investigate the function /(x) = 1 -x and show that ||g 2 -/|| >\ for every geC and that ^ cannot be improved.

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“…First we prove the aforementioned result [1] concerning extending a given function to a square. (g~\x)) whenever x e g (F).…”

Section: Elementary Observationsmentioning

confidence: 89%

“…Let C denote the space of continuous functions mapping [0,1] into itself and endowed with the sup metric. It has been shown that C 2 = {f°f:fe C} is an analytic but non-Borel subset of C. This implies that there is no simple geometric characterization for a function being a square.…”

mentioning

confidence: 99%

Approximations of continuous functions by squares

Humke

Laczkovich

1990

Ergod. Th. Dynam. Sys.

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“…However, the paper [44] by Humke and Laczkowich shows that this set is analytic and non-Borel in C(I, I). Nonexistence of roots is typical also for C 1 -smooth functions with two fixed points which was observed by Weinian Zhang in [182].…”

Section: Vol 87 (2014) Recent Results On Iteration Theory 217mentioning

confidence: 99%

Recent results on iteration theory: iteration groups and semigroups in the real case

Zdun

Solarz

2013

Aequat. Math.

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Abstract. In this survey paper we present some recent results in iteration theory. Mainly, we focus on the problems concerning real iteration groups (flows) and semigroups (semiflows) such as existence, regularity and embeddability. We also discuss some issues associated to the problem of embeddability, i.e. iterative roots and approximate iterability. The topics of this paper are: (1) measurable iteration semigroups; (2) embedding of diffeomorphisms in regular iteration semigroups in the space R n ; (3) iteration groups of fixed point free homeomorphisms on the plane; (4) embedding of interval homeomorphisms with two fixed points in a regular iteration group; (5) commuting functions and embeddability; (6) iterative roots; (7) the structure of iteration groups on an interval; (8) iteration groups of homeomorphisms of the circle; (9) approximately iterable functions; (10) set-valued iteration semigroups; (11) iterations of mean-type mappings; (12) Hayers-Ulam stability of the translation equation. Most of the results presented here were obtained by means of functional equations. We indicate the relations between iteration theory and functional equations.

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“…And this concerns both the purely set-theoretic case (cf., for instance, [75] by G. Zimmermann, but also [41] by R. E. Rice, B. Schweizer and A. Sklar) and the case of regular mappings, no matter we have in mind the continuity (see [11] by P.D. Humke and M. Laczkovich, [45] by K. Simon, and [4] by A.M. Blokh), or the holomorphicity (cf.…”

Section: Multifunction Casementioning

confidence: 99%

Recent results on iterative roots

Jarczyk

2014

ESAIM: Proc.

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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.

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The Borel structure of iterates of continuous functions

Abstract: The set of increasing (= nondecreasing) and decreasing (= nonincreasing) functions in % will be denoted by J and 2, respectively.

Cited by 24 publications

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Approximations of continuous functions by squares

Approximations of continuous functions by squares

Recent results on iteration theory: iteration groups and semigroups in the real case

Recent results on iterative roots

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