Metric entropy for set-valued maps

Abstract

<p style='text-indent:20px;'>In this article we define a notion of metric entropy for an invariant measure associated to a set-valued map <inline-formula><tex-math id="M1">\begin{document}$ F $\end{document}</tex-math></inline-formula> on a compact metric space <inline-formula><tex-math id="M2">\begin{document}$ X $\end{document}</tex-math></inline-formula>. Besides, we describe its main properties and prove the <i>Half Variational Principle</i>, which relates the metric entropy with the notion of topological entropy given in [<xref ref-type="bibr" rid="b13">13</xref>] for this class of maps.</p>

Equidistribution for matings of quadratic maps with the modular group

“…In later work [12], we prove that the measures and maximize entropy: the metric entropy in [31] yields equality for the half-variational principle with the topological entropy in [17].…”

Topological entropy of multivalued maps in topological spaces and hyperspaces

“…[2,3,6,7,8,15,16,19,27,31,35]), the results for metric entropy are still quite rare (see e.g. [14,28,33]).…”

“…In a multivalued case, as far as we know, there is only an analogous inequality to the one of Goodwyn in [33,Theorem 3.2], where the topological entropy is understood in the sense of [27]. The reverse inequality is examined under certain special restrictions in [33,Theorem 3.7] and in [14,Theorem 1.4], where the topological entropy is considered this time in the sense of [15].…”

“…At first, we will recall the related definitions of metric and topological entropy for multivalued maps and the "half variational principles", as it was presented in [33]. Since the standard Adler-Konheim-McAndrew type definitions (cf.…”

“…[13,17]) for single-valued continuous maps on compact metric spaces can be regarded as particular cases of their multivalued extensions, we will omit them here. Then we will show the counter-examples demonstrating the non-validity of the arguments applied to principles in [33]. Finally, we will present with comments their particular corrected versions and especially formulated the full variational principle for special lower semicontinuous maps with convex, compact values in a compact subset of a Banach space.…”