On fractional iterates of a homeomorphism of the plane

Leśniak, Zbigniew

doi:10.4064/ap79-2-4

Cited by 15 publications

(15 citation statements)

References 6 publications

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“…(1) we obtain that g is a bijective map. [19].) If f is a homeomorphism of R 2 onto itself, g is continuous and g n = f for some positive integer n, then g is also a homeomorphism of R 2 onto itself.…”

Section: Invariance Of Equivalence Classesmentioning

confidence: 99%

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On fractional iterates of a Brouwer homeomorphism embeddable in a flow

Leśniak

2010

Journal of Mathematical Analysis and Applications

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We present a method for finding continuous (and consequently homeomorphic) orientation preserving iterative roots of a Brouwer homeomorphism which is embeddable in a flow.To obtain the roots we use a countable family of maximal parallelizable regions of the flow which is a cover of the plane. The maximal parallelizable regions are unions of equivalence classes of an appropriate equivalence relation. We show that if an equivalence class is invariant under the nth iterate of a Brouwer homeomorphism g, then it is invariant under g. We use this fact to prove that each maximal parallelizable region of the flow must be invariant under all homeomorphic orientation preserving iterative roots of the given Brouwer homeomorphism.

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Section: Invariance Of Equivalence Classesmentioning

confidence: 99%

“…Moreover, if f has no fixed points, then g has no fixed points. [19].) Let f be a homeomorphism of R 2 onto itself which preserves orientation.…”

Section: Invariance Of Equivalence Classesmentioning

confidence: 99%

On fractional iterates of a Brouwer homeomorphism embeddable in a flow

Leśniak

2010

Journal of Mathematical Analysis and Applications

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“…Many advances have been made to iterative roots for various kinds of maps, e.g., continuous strictly monotone self-maps of intervals [3,6], homeomorphisms of the circle [28,29,34], series and transseries [4], set-valued functions [9,18,24,25], high dimensional mappings [15][16][17]23]. Many researchers are also devoted to some properties of iterative roots, for instance, differentiability [11,13,33,38], approximation [39], stability [40,41], category and measure [2,27].…”

Section: Introductionmentioning

confidence: 99%

Extension of iterative roots

Shi

Chen

2013

Aequat. Math.

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“…The same paper shows also that even in the class of piecewise linear functions equation (2.1) leads to non-trivial questions. Note also the paper [166] by Z. Leśniak which contains a construction of continuous roots of Sperner homeomorphisms of the plane.…”

Section: Theorem 25 the N-thčebyšev Polynomial Has A Square Root Ifmentioning

confidence: 99%