On the Group of Piecewise Linear Monotone Bijections of an Arc.

Ulrich Abel

Displaying similar documents to “On the Group of Piecewise Linear Monotone Bijections of an Arc.”

On Whitney pairs

Marianna Csörnyei (1999)

Fundamenta Mathematicae

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A simple arc ϕ is said to be a Whitney arc if there exists a non-constant function f such that $l i m_{x \mapsto x_{0}} (| f (x) - f (x_{0}) |) / (| ϕ (x) - ϕ (x_{0}) |) = 0$ for every $x_{0}$ . G. Petruska raised the question whether there exists a simple arc ϕ for which every subarc is a Whitney arc, but for which there is no parametrization satisfying $l i m_{t \mapsto t_{0}} (| t - t_{0} |) / (| ϕ (t) - ϕ (t_{0}) |) = 0$ . We answer this question partially, and study the structural properties of possible monotone, strictly monotone and VBG* functions f and associated Whitney arcs.

Über monotone Matrixfunktionen

K.T'. Löwner (1934)

Mathematische Zeitschrift

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Characterizing the arc by composition of functions

Charles L. Hagopian (1975)

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Whitney arcs and 1-critical arcs

Marianna Csörnyei, Jan Kališ, Luděk Zajíček (2008)

Fundamenta Mathematicae

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A simple arc γ ⊂ ℝⁿ is called a Whitney arc if there exists a non-constant real function f on γ such that $l i m_{y \to x, y \in γ} | f (y) - f (x) | / | y - x | = 0$ for every x ∈ γ; γ is 1-critical if there exists an f ∈ C¹(ℝⁿ) such that f’(x) = 0 for every x ∈ γ and f is not constant on γ. We show that the two notions are equivalent if γ is a quasiarc, but for general simple arcs the Whitney property is weaker. Our example also gives an arc γ in ℝ² each of whose subarcs is a monotone Whitney arc, but which is not a strictly monotone Whitney...