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

On Whitney pairs

Marianna Csörnyei (1999)

Fundamenta Mathematicae

Similarity:

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 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 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.

Whitney arcs and 1-critical arcs

Marianna Csörnyei, Jan Kališ, Luděk Zajíček (2008)

Fundamenta Mathematicae

Similarity:

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 x , y 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...

Monotone homogeneity of dendrites

Janusz Jerzy Charatonik, Włodzimierz J. Charatonik (1997)

Commentationes Mathematicae Universitatis Carolinae

Similarity:

Sufficient as well as necessary conditions are studied for a dendrite or a dendroid to be homogeneous with respect to monotone mappings. The obtained results extend ones due to H. Kato and the first named author. A number of open problems are asked.