1,054 research outputs found
Weak expansiveness for actions of sofic groups
In this paper, we shall introduce -expansiveness and asymptotical
-expansiveness for actions of sofic groups. By the definitions, each
-expansive action of sofic groups is asymptotically -expansive. We show
that each expansive action of sofic groups is -expansive, and, for any given
asymptotically -expansive action of sofic groups, the entropy function (with
respect to measures) is upper semi-continuous and hence the system admits a
measure with maximal entropy.
Observe that asymptotically -expansive property was firstly introduced and
studied by Misiurewicz for -actions using the language of
topological conditional entropy. And thus in the remaining part of the paper,
we shall compare our definitions of weak expansiveness for actions of sofic
groups with the definitions given in the same spirit of Misiurewicz's ideas
when the group is amenable. It turns out that these two definitions are
equivalent in this setting.Comment: to appear in Journal of Functional Analysi
Discrete dynamical systems in group theory
In this expository paper we describe an unifying approach for many known
entropies in Mathematics. First we recall the notion of semigroup entropy h_S
in the category S of normed semigroups and contractive homomorphisms, recalling
also its properties. For a specific category X and a functor F from X to S, we
have the entropy h_F, defined by the composition of h_S with F, which
automatically satisfies the same properties proved for h_S. This general scheme
permits to obtain many of the known entropies as h_F, for appropriately chosen
categories X and functors F. In the last part we recall the definition and the
fundamental properties of the algebraic entropy for group endomorphisms, noting
how its deeper properties depend on the specific setting. Finally we discuss
the notion of growth for flows of groups, comparing it with the classical
notion of growth for finitely generated groups
Topological Entropy and Algebraic Entropy for group endomorphisms
The notion of entropy appears in many fields and this paper is a survey about
entropies in several branches of Mathematics. We are mainly concerned with the
topological and the algebraic entropy in the context of continuous
endomorphisms of locally compact groups, paying special attention to the case
of compact and discrete groups respectively. The basic properties of these
entropies, as well as many examples, are recalled. Also new entropy functions
are proposed, as well as generalizations of several known definitions and
results. Furthermore we give some connections with other topics in Mathematics
as Mahler measure and Lehmer Problem from Number Theory, and the growth rate of
groups and Milnor Problem from Geometric Group Theory. Most of the results are
covered by complete proofs or references to appropriate sources
The mathematical research of William Parry FRS
In this article we survey the mathematical research of the late William (Bill) Parry, FRS
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