PM functions, their characteristic intervals and iterative roots

Zhang, Weinian

doi:10.4064/ap-65-2-119-128

Cited by 65 publications

(48 citation statements)

References 7 publications

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“…Now we are interested in the more complicated case H(F ) ≥ 2. Then, as Weinian Zhang proved in [72], the function F has no continuous iterative roots of order r greater than #S(F ). So the problem is: Does F ∈ PM(a, b) with H(F ) ≥ 2 have an iterative root of order r for r ≤ #S(F )?…”

Section: In 1944mentioning

confidence: 99%

“…[72] by Weinian Zhang and [25] by Lin Li, Dilian Yang and Weinian Zhang) and can be reduced to the monotonic case in the following way. By Fact 1.3 we know that the function F is not monotonic on the interval [a, b] but it is monotonic on the interval…”

Section: In 1944mentioning

confidence: 99%

“…This can be extended on [a, b] to a continuous root of F . The method of determining iterative roots of piecewise monotonic functions by using their characteristic intervals was proposed and described in Chinese by Jingzhong Zhang and Lu Yang [68] in 1983, and then improved by Weinian Zhang [72] in 1997. Investigation of this method in the case H(F ) = 1 has been recently completed by Liu Liu and Weinian Zhang in the paper [27].…”

Section: In 1944mentioning

confidence: 99%

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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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Section: In 1944mentioning

confidence: 99%

Section: In 1944mentioning

confidence: 99%

Section: In 1944mentioning

confidence: 99%

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Recent results on iterative roots

Jarczyk

2014

ESAIM: Proc.

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“…Their method based on the notion of so called "characteristic interval" was described in the paper mentioned above and then was developed in [85,183].…”

Section: Theorem 24 ([14]) Let F Be a Strict Horseshoe Map Of Type (mentioning

confidence: 99%

“…By PM (I) we denote the set of all continuous piecewise monotonic functions with finitely many forts. If [84,183]). For a given function F ∈ PM(I) we define the nonmonotonicity height H(F ) as the least k ≥ 0 satisfying S(F k ) = S(F k+1 ) if such a k exists and ∞ otherwise.…”

Section: Theorem 24 ([14]) Let F Be a Strict Horseshoe Map Of Type (mentioning

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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Recent Developments in the Translation Equation and Its Stability

Przebieracz

2017

Developments in Functional Equations and Related Topics

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PM functions, their characteristic intervals and iterative roots

Abstract: Abstract. The concept of characteristic interval for piecewise monotone functions is introduced and used in the study of their iterative roots on a closed interval.

Cited by 65 publications

References 7 publications

Recent results on iterative roots

Recent results on iterative roots

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

Recent Developments in the Translation Equation and Its Stability

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