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Some cohomological aspects of the Banach fixed point principle

Ludvík Janoš (2011)

Mathematica Bohemica

Let T : X → X be a continuous selfmap of a compact metrizable space X . We prove the equivalence of the following two statements: (1) The mapping T is a Banach contraction relative to some compatible metric on X . (2) There is a countable point separating family ℱ ⊂ 𝒞 ( X ) of non-negative functions f ∈ 𝒞 ( X ) such that for every f ∈ ℱ there is g ∈ 𝒞 ( X ) with f = g - g ∘ T .

Some constructions of strictly ergodic non-regular Toeplitz flows

A. Iwanik, Y. Lacroix (1994)

Studia Mathematica

We give a necessary and sufficient condition for a Toeplitz flow to be strictly ergodic. Next we show that the regularity of a Toeplitz flow is not a topological invariant and define the "eventual regularity" as a sequence; its behavior at infinity is topologically invariant. A relation between regularity and topological entropy is given. Finally, we construct strictly ergodic Toeplitz flows with "good" cyclic approximation and non-discrete spectrum.

Some dynamical properties of S-unimodal maps

Tomasz Nowicki (1993)

Fundamenta Mathematicae

We study 1) the slopes of central branches of iterates of S-unimodal maps, comparing them to the derivatives on the critical trajectory, 2) the hyperbolic structure of Collet-Eckmann maps estimating the exponents, and under a summability condition 3) the images of the density one under the iterates of the Perron-Frobenius operator, 4) the density of the absolutely continuous invariant measure.

Some families of pseudo-processes

J. Kłapyta (1994)

Annales Polonici Mathematici

We introduce several types of notions of dis persive, completely unstable, Poisson unstable and Lagrange uns table pseudo-processes. We try to answer the question of how many (in the sense of Baire category) pseudo-processes with each of these properties can be defined on the space ℝ m . The connections are discussed between several types of pseudo-processes and their limit sets, prolongations and prolongational limit sets. We also present examples of applications of the above results to pseudo-processes...

Some remarks about strong proximality of compact flows

A. Bouziad, J.-P. Troallic (2009)

Colloquium Mathematicae

This note aims at providing some information about the concept of a strongly proximal compact transformation semigroup. In the affine case, a unified approach to some known results is given. It is also pointed out that a compact flow (X,𝓢) is strongly proximal if (and only if) it is proximal and every point of X has an 𝓢-strongly proximal neighborhood in X. An essential ingredient, in the affine as well as in the nonaffine case, turns out to be the existence of a unique minimal subset.

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