3,335 research outputs found
Infinite words and universal free actions
This is the second paper in a series of three, where we take on the unified
theory of non-Archimedean group actions, length functions and infinite words.
Here, for an arbitrary group of infinite words over an ordered abelian
group we construct a -tree equipped with a free
action of . Moreover, we show that is a universal tree for in
the sense that it isometrically embeds in every -tree equipped with a
free -action compatible with the original length function on .Comment: 20 pages, 4 figure
The Non-Archimedean Theory of Discrete Systems
In the paper, we study behavior of discrete dynamical systems (automata)
w.r.t. transitivity; that is, speaking loosely, we consider how diverse may be
behavior of the system w.r.t. variety of word transformations performed by the
system: We call a system completely transitive if, given arbitrary pair
of finite words that have equal lengths, the system , while
evolution during (discrete) time, at a certain moment transforms into .
To every system , we put into a correspondence a family of continuous maps of a suitable non-Archimedean metric space
and show that the system is completely transitive if and only if the family
is ergodic w.r.t. the Haar measure; then we find
easy-to-verify conditions the system must satisfy to be completely transitive.
The theory can be applied to analyze behavior of straight-line computer
programs (in particular, pseudo-random number generators that are used in
cryptography and simulations) since basic CPU instructions (both numerical and
logical) can be considered as continuous maps of a (non-Archimedean) metric
space of 2-adic integers.Comment: The extended version of the talk given at MACIS-201
Affine actions on non-archimedean trees
We initiate the study of affine actions of groups on -trees for a
general ordered abelian group ; these are actions by dilations rather
than isometries. This gives a common generalisation of isometric action on a
-tree, and affine action on an -tree as studied by I. Liousse. The
duality between based length functions and actions on -trees is
generalised to this setting. We are led to consider a new class of groups:
those that admit a free affine action on a -tree for some .
Examples of such groups are presented, including soluble Baumslag-Solitar
groups and the discrete Heisenberg group.Comment: 27 pages. Section 1.4 expanded, typos corrected from previous versio
-free groups are CAT(0)
We show that every group with free -length function is CAT(0).Comment: To be published in the Journal of the London Mathematical Society.
This version is very close to the accepted version. The exposition greatly
improved due to the referee's comment
New perspectives in Arakelov geometry
In this survey, written for the proceedings of the VII meeting of the CNTA
held in May 2002 in Montreal, we describe how Connes' theory of spectral
triples provides a unified view, via noncommutative geometry, of the
archimedean and the totally split degenerate fibers of an arithmetic surface.Comment: 20 pages, 10pt LaTeX, 2 eps figures (v3: some changes for the final
version
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