6,515 research outputs found
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
Preperiodic points and unlikely intersections
In this article, we combine complex-analytic and arithmetic tools to study
the preperiodic points of one-dimensional complex dynamical systems. We show
that for any fixed complex numbers a and b, and any integer d at least 2, the
set of complex numbers c for which both a and b are preperiodic for z^d+c is
infinite if and only if a^d = b^d. This provides an affirmative answer to a
question of Zannier, which itself arose from questions of Masser concerning
simultaneous torsion sections on families of elliptic curves. Using similar
techniques, we prove that if two complex rational functions f and g have
infinitely many preperiodic points in common, then they must have the same
Julia set. This generalizes a theorem of Mimar, who established the same result
assuming that f and g are defined over an algebraic extension of the rationals.
The main arithmetic ingredient in the proofs is an adelic equidistribution
theorem for preperiodic points over number fields and function fields, with
non-archimedean Berkovich spaces playing an essential role.Comment: 26 pages. v3: Final version to appear in Duke Math.
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