Past Analytic Topology in Mathematics and Computer Science

11 March 2015
16:00
Dr Good
Abstract

Abstract: Joint work with Syahida Che Dzul-Kifli

 

Let $f:X\to X$ be a continuous function on a compact metric space forming a discrete dynamical system. There are many definitions that try to capture what it means for the function $f$ to be chaotic. Devaney’s definition, perhaps the most frequently cited, asks for the function $f$ to be topologically transitive, have a dense set of periodic points and is sensitive to initial conditions.  Bank’s et al show that sensitive dependence follows from the other two conditions and Velleman and Berglund show that a transitive interval map has a dense set of periodic points.  Li and Yorke (who coined the term chaos) show that for interval maps, period three implies chaos, i.e. that the existence of a period three point (indeed of any point with period having an odd factor) is chaotic in the sense that it has an uncountable scrambled set.

 

The existence of a period three point is In this talk we examine the relationship between transitivity and dense periodic points and look for simple conditions that imply chaos in interval maps. Our results are entirely elementary, calling on little more than the intermediate value theorem.

  • Analytic Topology in Mathematics and Computer Science
18 February 2015
16:00
Abstract
<pre wrap="">Compact F-spaces play an important role in the area of compactification theory, the prototype being w*, the Stone-Cech remainder of the integers. Two curious topological characteristics of compact F-spaces are that they don’t contain convergent sequences (apart from the constant ones), and moreover, that they often contain points that don’t lie in the boundary of any countable subset (so-called weak P-points). In this talk we investigate the space of self-maps S(X) on compact zero-dimensional F-spaces X, endowed with the compact-open topology. A natural question is whether S(X) reflects properties of the ground space X. Our main result is that for zero-dimensional compact F-spaces X, also S(X) doesn’t contain convergent sequences. If time permits, I will also comment on the existence of weak P-points in S(X). This is joint work with Richard Lupton.</pre>
  • Analytic Topology in Mathematics and Computer Science
26 November 2014
16:00
Leobardo Fernández Román
Abstract
A continuum is a non-empty
compact connected metric space.
Given a continuum X let P(X) be the
power set of X. We define the following
set functions:
 
T:P(X) to P(X) given by, for each A in P(X),
T(A) = X \ { x in X : there is a continuum W
such that x is in Int(W) and W does not
intersect A}.
 
K:P(X) to P(X) given by, for each A in P(X)
K(A) = Intersection{ W : W is a subcontinuum
of X and A is in the interior of W}.
 
Also, it is possible to define the arcwise
connected version of these functions.
Given an arcwise connected continuum X:
 
Ta:P(X) to P(X) given by, for each A in P(X),
Ta(A) = X \ { x in X : there is an arcwise
connected continuum W such that x is in
Int(W) and W does not intersect A}.
 
Ka:P(X) to P(X) given by, for each A in P(X),
Ka(A) = Intersection{ W : W is an arcwise
connected subcontinuum of X and A is in
the interior of W}
 
Some properties, examples and relations
between these functions are going to be
presented.
  • Analytic Topology in Mathematics and Computer Science

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