9,150 research outputs found
Topological entropy for automorphisms of totally disconnected locally compact groups
We give a "limit-free formula" simplifying the computation of the topological
entropy for topological automorphisms of totally disconnected locally compact
groups. This result allows us to extend several basic properties of the
topological entropy known to hold for compact groups
Extremal k-pseudocompact abelian groups
For a cardinal k, generalizing a recent result of Comfort and van Mill, we
prove that every k-pseudocompact abelian group of weight >k has some proper
dense k-pseudocompact subgroup and admits some strictly finer k-pseudocompact
group topology.Comment: 24 page
Adjoint entropy vs Topological entropy
Recently the adjoint algebraic entropy of endomorphisms of abelian groups was
introduced and studied. We generalize the notion of adjoint entropy to
continuous endomorphisms of topological abelian groups. Indeed, the adjoint
algebraic entropy is defined using the family of all finite-index subgroups,
while we take only the subfamily of all open finite-index subgroups to define
the topological adjoint entropy. This allows us to compare the (topological)
adjoint entropy with the known topological entropy of continuous endomorphisms
of compact abelian groups. In particular, the topological adjoint entropy and
the topological entropy coincide on continuous endomorphisms of totally
disconnected compact abelian groups. Moreover, we prove two Bridge Theorems
between the topological adjoint entropy and the algebraic entropy using
respectively the Pontryagin duality and the precompact duality.Comment: 18 page
Functorial topologies and finite-index subgroups of abelian groups
In the general context of functorial topologies, we prove that in the lattice
of all group topologies on an abelian group, the infimum between the Bohr
topology and the natural topology is the profinite topology. The profinite
topology and its connection to other functorial topologies is the main
objective of the paper. We are particularly interested in the poset C(G) of all
finite-index subgroups of an abelian group G, since it is a local base for the
profinite topology of G. We describe various features of the poset C(G) (its
cardinality, its cofinality, etc.) and we characterize the abelian groups G for
which C(G)\{G} is cofinal in the poset of all subgroups of G ordered by
inclusion. Finally, for pairs of functorial topologies T, S we define the
equalizer E(T,S), which permits to describe relevant classes of abelian groups
in terms of functorial topologies.Comment: 19 page
The Bridge Theorem for totally disconnected LCA groups
For a totally disconnected locally compact abelian group, we prove that the
topological entropy of a continuous endomorphism coincides with the algebraic
entropy of the dual endomorphism with respect to the Pontryagin duality.
Moreover, this result is extended to all locally compact abelian groups under
the assumption of additivity with respect to some fully invariant subgroups for
both the topological and the algebraic entropy
Topological entropy in totally disconnected locally compact groups
Let be a topological group, let be a continuous endomorphism of
and let be a closed -invariant subgroup of . We study whether
the topological entropy is an additive invariant, that is,
where
is the map induced by . We concentrate on the case
when is locally compact totally disconnected and is either compact or
normal. Under these hypotheses, we show that the above additivity property
holds true whenever and . As an application we
give a dynamical interpretation of the scale , by showing that is the topological entropy of a suitable map induced by .
Finally, we give necessary and sufficient conditions for the equality to hold.Comment: 18 page
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
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