### A chaotic function with zero topological entropy having a non-perfect attractor

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We classify the braid types that can occur for finite unions of periodic orbits of diffeomorphisms of surfaces of genus one with zero topological entropy.

The Lefschetz zeta function associated to a continuous self-map f of a compact manifold is a rational function P/Q. According to the parity of the degrees of the polynomials P and Q, we analyze when the set of periodic points of f is infinite and when the topological entropy is positive.

In this paper we study the commutativity property for topological sequence entropy. We prove that if $X$ is a compact metric space and $f,g:X\to X$ are continuous maps then ${h}_{A}(f\circ g)={h}_{A}(g\circ f)$ for every increasing sequence $A$ if $X=[0,1]$, and construct a counterexample for the general case. In the interim, we also show that the equality ${h}_{A}\left(f\right)={h}_{A}(f{|}_{{\cap}_{n\ge 0}{f}^{n}\left(X\right)})$ is true if $X=[0,1]$ but does not necessarily hold if $X$ is an arbitrary compact metric space.

Modifying Bowen's entropy, we introduce a new uniform entropy. We prove that the completion theorem for uniform entropy holds in the class of all metric spaces. However, the completion theorem for Bowen's entropy does not hold in the class of all totally bounded metric spaces.

Approximation theory in the context of probability density function turns out to go beyond the classical idea of orthogonal projection. Special tools have to be designed so as to respect the nonnegativity of the approximate function. We develop here and justify from the theoretical point of view an approximation procedure introduced by Levermore [Levermore, J. Stat. Phys. 83 (1996) 1021–1065] and based on an entropy minimization principle under moment constraints. We prove in particular a global...

Approximation theory in the context of probability density function turns out to go beyond the classical idea of orthogonal projection. Special tools have to be designed so as to respect the nonnegativity of the approximate function. We develop here and justify from the theoretical point of view an approximation procedure introduced by Levermore [Levermore, J. Stat. Phys.83 (1996) 1021–1065] and based on an entropy minimization principle under moment constraints. We prove in particular...

For mappings $f\phantom{\rule{0.166667em}{0ex}}:\phantom{\rule{0.166667em}{0ex}}S\to S$, where $S$ is a merotopic space equipped with a diameter function, we introduce and examine an entropy, called the $\delta $-entropy. The topological entropy and the entropy of self-mappings of metric spaces are shown to be special cases of the $\delta $-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 $\delta $-entropy of $f\phantom{\rule{0.166667em}{0ex}}:\phantom{\rule{0.166667em}{0ex}}S\to S$.