7,492 research outputs found
Dimension of graphoids of rational vector-functions
Let be a countable family of rational functions of two variables with
real coefficients. Each rational function can be thought as a
continuous function taking values in the projective line
and defined on a cofinite subset of the torus
. Then the family \F determines a continuous vector-function
defined on the dense -set of . The closure of its graph
in is called the
{\em graphoid} of the family . We prove the graphoid has
topological dimension . If the family contains all
linear fractional transformations for ,
then the graphoid has cohomological dimension
for any non-trivial 2-divisible abelian group .
Hence the space is a natural example of a compact space that is
not dimensionally full-valued and by this property resembles the famous
Pontryagin surface.Comment: 20 page
Continuity argument revisited: geometry of root clustering via symmetric products
We study the spaces of polynomials stratified into the sets of polynomial
with fixed number of roots inside certain semialgebraic region , on its
border, and at the complement to its closure. Presented approach is a
generalisation, unification and development of several classical approaches to
stability problems in control theory: root clustering (-stability) developed
by R.E. Kalman, B.R. Barmish, S. Gutman et al., -decomposition(Yu.I.
Neimark, B.T. Polyak, E.N. Gryazina) and universal parameter space method(A.
Fam, J. Meditch, J.Ackermann).
Our approach is based on the interpretation of correspondence between roots
and coefficients of a polynomial as a symmetric product morphism.
We describe the topology of strata up to homotopy equivalence and, for many
important cases, up to homeomorphism. Adjacencies between strata are also
described. Moreover, we provide an explanation for the special position of
classical stability problems: Hurwitz stability, Schur stability,
hyperbolicity.Comment: 45 pages, 4 figure
Topology of definable Hausdorff limits
Let be a set definable in an o-minimal expansion of the
real field, be its projection, and assume that the non-empty
fibers are compact for all and uniformly bounded,
{\em i.e.} all fibers are contained in a ball of fixed radius If
is the Hausdorff limit of a sequence of fibers we give an
upper-bound for the Betti numbers in terms of definable sets
explicitly constructed from a fiber In particular, this allows to
establish effective complexity bounds in the semialgebraic case and in the
Pfaffian case. In the Pfaffian setting, Gabrielov introduced the {\em relative
closure} to construct the o-minimal structure \S_\pfaff generated by Pfaffian
functions in a way that is adapted to complexity problems. Our results can be
used to estimate the Betti numbers of a relative closure in the
special case where is empty.Comment: Latex, 23 pages, no figures. v2: Many changes in the exposition and
notations in an attempt to be clearer, references adde
Entropy in Dimension One
This paper completely classifies which numbers arise as the topological
entropy associated to postcritically finite self-maps of the unit interval.
Specifically, a positive real number h is the topological entropy of a
postcritically finite self-map of the unit interval if and only if exp(h) is an
algebraic integer that is at least as large as the absolute value of any of the
conjugates of exp(h); that is, if exp(h) is a weak Perron number. The
postcritically finite map may be chosen to be a polynomial all of whose
critical points are in the interval (0,1). This paper also proves that the weak
Perron numbers are precisely the numbers that arise as exp(h), where h is the
topological entropy associated to ergodic train track representatives of outer
automorphisms of a free group.Comment: 38 pages, 15 figures. This paper was completed by the author before
his death, and was uploaded by Dylan Thurston. A version including endnotes
by John Milnor will appear in the proceedings of the Banff conference on
Frontiers in Complex Dynamic
Lectures on Klein surfaces and their fundamental group
The goal of these lectures is to give an introduction to the study of the
fundamental group of a Klein surface. We start by reviewing the topological
classification of Klein surfaces and by explaining the relation with real
algebraic curves. Then we introduce the fundamental group of a Klein surface
and present its main basic properties. Finally, we study the variety of unitary
representations of this group and relate it to the representation variety of
the topological fundamental group of the underlying Riemann surface.Comment: To appear in the collection Advanced Courses in Mathematics - CRM
Barcelon
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