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Entropies of self-mappings of topological spaces with richer structures

Miroslav Katětov — 1993

Commentationes Mathematicae Universitatis Carolinae

For mappings f : S S , where S is a merotopic space equipped with a diameter function, we introduce and examine an entropy, called the δ -entropy. The topological entropy and the entropy of self-mappings of metric spaces are shown to be special cases of the δ -entropy. Some connections with other characteristics of self-mappings are considered. We also introduce and examine an entropy for subsets of S N , which is closely connected with the δ -entropy of f : S S .

Entropy-like functionals: conceptual background and some results

Miroslav Katětov — 1992

Commentationes Mathematicae Universitatis Carolinae

We describe a conceptual approach which provides a unified view of various entropy-like functionals on the class of semimetric spaces, endowed with a bounded measure. The entropy E considered in the author’s previous articles is modified so as to assume finite values for a fairly wide class of spaces which fail to be totally bounded.

On entropy-like functionals and codes for metrized probability spaces II

Miroslav Katětov — 1992

Commentationes Mathematicae Universitatis Carolinae

In Part I, we have proved characterization theorems for entropy-like functionals δ , λ , E , Δ and Λ restricted to the class consisting of all finite spaces P 𝔚 , the class of all semimetric spaces equipped with a bounded measure. These theorems are now extended to the case of δ , λ and E defined on the whole of 𝔚 , and of Δ and Λ restricted to a certain fairly wide subclass of 𝔚 .

An approach to covering dimensions

Miroslav Katětov — 1995

Commentationes Mathematicae Universitatis Carolinae

Using certain ideas connected with the entropy theory, several kinds of dimensions are introduced for arbitrary topological spaces. Their properties are examined, in particular, for normal spaces and quasi-discrete ones. One of the considered dimensions coincides, on these spaces, with the Čech-Lebesgue dimension and the height dimension of posets, respectively.

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