Linear Li–Yorke Chaos in a Finite-Dimensional Space with Weak Topology

It is well known that a finite-dimensional linear system cannot be chaotic. In this article, by introducing a weak topology into a two-dimensional Euclidean space, it shows that Li–Yorke chaos can be generated by a linear map, where the weak topology is induced by a linear functional. Some examples of linear systems are presented, some are chaotic while some others regular. Consequently, several open problems are posted.

Ornstein-Uhlenbeck Semigroup on the dual space of Gelfand-Shilov Spaces of Beurling type

We use a previously obtained topological characterization of Gelfand-Shilov spaces of Beurling type to characterize its dual using Riesz representation theorem. Using the characterization of the dual space equipped with the weak topology, we study the action of Ornstein-Uhlenbeck Semigroup on the dual space.

Weak fuzzy topology on vector spaces

In this paper, we study the concept of weak linear fuzzy topology on a fuzzy topological vector space as a generalization of usual weak topology. We prove that this fuzzy topology consists of all weakly lower semi-continuous fuzzy sets on a given vector space when K (R or C) endowed with its usual fuzzy topology. In the case that the fuzzy topology of K is different from the usual fuzzy topology, we show that the weak fuzzy topology is not equivalent with the fuzzy topology of weakly lower semi-continuous fuzzy sets.

$ \Gamma $-convergence of quadratic functionals with non uniformly elliptic conductivity matrices

<p style='text-indent:20px;'>We investigate the homogenization through <inline-formula><tex-math id="M2">\begin{document}$ \Gamma $\end{document}</tex-math></inline-formula>-convergence for the <inline-formula><tex-math id="M3">\begin{document}$ L^2({\Omega}) $\end{document}</tex-math></inline-formula>-weak topology of the conductivity functional with a zero-order term where the matrix-valued conductivity is assumed to be non strongly elliptic. Under proper assumptions, we show that the homogenized matrix <inline-formula><tex-math id="M4">\begin{document}$ A^\ast $\end{document}</tex-math></inline-formula> is provided by the classical homogenization formula. We also give algebraic conditions for two and three dimensional <inline-formula><tex-math id="M5">\begin{document}$ 1 $\end{document}</tex-math></inline-formula>-periodic rank-one laminates such that the homogenization result holds. For this class of laminates, an explicit expression of <inline-formula><tex-math id="M6">\begin{document}$ A^\ast $\end{document}</tex-math></inline-formula> is provided which is a generalization of the classical laminate formula. We construct a two-dimensional counter-example which shows an anomalous asymptotic behaviour of the conductivity functional.</p>

Uniformly positive entropy of induced transformations

Let $(X,T)$ be a topological dynamical system consisting of a compact metric space X and a continuous surjective map $T : X \to X$ . By using local entropy theory, we prove that $(X,T)$ has uniformly positive entropy if and only if so does the induced system $({\mathcal {M}}(X),\widetilde {T})$ on the space of Borel probability measures endowed with the weak* topology. This result can be seen as a version for the notion of uniformly positive entropy of the corresponding result for topological entropy due to Glasner and Weiss.