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Metric entropy of convex hulls in Hilbert spaces

100%

Li W. V., Linde W.

Studia Mathematica

2000

tom 139

nr 1

29-45

Let T be a precompact subset of a Hilbert space. We estimate the metric entropy of co(T), the convex hull of T, by quantities originating in the theory of majorizing measures. In a similar way, estimates of the Gelfand width are provided. As an application we get upper bounds for the entropy of co(T), $T={t_1,t_2,...}$, $||t_j||≤a_j$, by functions of the $a_j$'s only. This partially answers a question raised by K. Ball and A. Pajor (cf. [1]). Our estimates turn out to be optimal in the case of slowly decreasing sequences $(a_j)_{j=1}^∞$.

Metric Entropy of Nonautonomous Dynamical Systems

100%

Kawan C.

Nonautonomous Dynamical Systems

2014

tom 1

We introduce the notion of metric entropy for a nonautonomous dynamical system given by a sequence (Xn, μn) of probability spaces and a sequence of measurable maps fn : Xn → Xn+1 with fnμn = μn+1. This notion generalizes the classical concept of metric entropy established by Kolmogorov and Sinai, and is related via a variational inequality to the topological entropy of nonautonomous systems as defined by Kolyada, Misiurewicz, and Snoha. Moreover, it shares several properties with the classical notion of metric entropy. In particular, invariance with respect to appropriately defined isomorphisms, a power rule, and a Rokhlin-type inequality are proved.

Ograniczanie wyników

1 Nonautonomous Dynamical Systems

1 Studia Mathematica

1 Kawan C.

1 Li W. V.

1 Linde W.

1 2013

1 2000

Wyniki wyszukiwania

Metric entropy of convex hulls in Hilbert spaces

Metric Entropy of Nonautonomous Dynamical Systems