Dynamical Systems Seminars (Friday 29 January)

[math]

MATHEMATICS SEMINARS 2010/DYNAMICAL SYSTEMS

Friday, 29 January

Venue: ICTP, Leonardo Building, EULER LECTURE HALL (terrace level)

14.00 G. Panti (Udine)
15.00 Break
15.30 G. Gentili (Firenze)
16.30 F. Vlacci (Firenze)

G. Panti
Kakutani-von Neumann maps on simplexes

Abstract: A Kakutani-von Neumann map is the push-forward of the group 
rotation (Z_2,+1) to a unit simplex via an appropriate topological 
quotient. The usual quotient towards the unit interval is given by the 
base 2 expansion of real numbers, which in turn is induced by the 
doubling map.

We replace the doubling map with an n-dimensional generalization of the 
tent map; this allows us to define Kakutani-von Neumann transformations 
in simplexes of arbitrary dimensions. The resulting maps are 
piecewise-linear bijections (not just mod 0 bijections), whose orbits 
are all uniformly distributed; in particular, they are uniquely ergodic 
w.r.t. the Lebesgue measure.

The forward orbit of a certain vertex provides an enumeration of all 
points in the simplex having dyadic coordinates, and this enumeration 
can be translated via the n-dimensional Minkowski function to an 
enumeration of all rational points. In the course of establishing the 
above results, we introduce a family of {+1,-1}-valued functions, 
constituting an n-dimensional analogue of the classical Walsh functions.
Paper reference:
http://arxiv.org/abs/1001.3324

                       * * * * *
G. Gentili
"Basic results on regular functions over quaternions"

Abstract. The richness of the theory of holomorphic functions of one 
complex variable arouse interest in  "resembling" theories for 
quaternionic functions of a quaternionic variable. The first aim of the 
talk is to present the definition of (Cullen or) slice regular function 
over quaternions, accompanied by  the main reasons that motivated it. 
The basic results of this recent theory will then be illustrated, paying 
  particular attention to those topics in which the results are diverse 
from those which hold true for the theory of complex holomorphic functions.
                       * * * * *
F. Vlacci
"Properties of zero sets of regular functions"

Abstract: The aim of this talk is to give a geometrical description of 
the zero sets of regular functions over quaternions; in particular we 
will focus our attention on the zero sets of regular polynomials and 
give a proof of the Fundamental Theorem of Algebra in this setting.
Some related topics will be then presented in order to show what kind of 
difficulties or open problems are still under investigation.