2.00-2.45. Kostas Karamenos, Brussels
TITLE: On the arithmetic nature of various constants at the accumulation points of unimodal maps and in particular of the logistic map
ABSTRACT: We have recently proposed a construction based on the symbolic dynamics of the class of unimodal maps, where the phenomenon of (week) chaos can be mapped in the non-normal (and possibly transcendental) character of some relevant constants. In some cases one can show explicitly that the corresponding constants are transcendental. We analyze these cases. We also explain a more recent result based on the statistical analysis of their binary expansions, that it could be seriously envisaged that the Feigenbaum constants \alpha and \delta for the logistic map are also transcendental. We also show a way to attack the same problem for the control parameter value at the first accumulation point. Finally, we analyze the B4 point of the logistic map thus confirming two relevant conjectures by Bailey and Broadhurst.
3.00-3.45. Heinz Bauschke, Mathematics and Statistics, University of Guelph
TITLE: Hundal's alternating projections counterexample and the proximal point algorithm
ABSTRACT: Hein Hundal recently constructed an alternating projections iteration that converges weakly but not in norm. In this talk, I will show how Hundal's example can be viewed as a sequence, generated by the proximal point algorithm, that converges weakly but not in norm. The existence of such a sequence was first established by Osman Guler; however, the construction proposed here is much simpler.
This is based on joint works with J. Burke, F. Deutsch, H. Hundal, E. Matouskova, S. Reich, and J. Vanderwerff.
4.00-4.45. Ivo Kortezov, Sofia
TITLE: Norm continuity of weakly continuous mappings into Banach spaces
ABSTRACT: A mapping f: Z\to X is called quasicontinuous (qc), if for any z in Z and open subset U of X, provided f(z)\in U, there is an open subset V of Z having z in its closure, such that f(V) is a subset of U. Such f has a lot of points of continuity if X is nice, but otherwise may fail to have a single one (e.g. X=Sorgenfrey line, Z=R, f=id). Using the non-existence of a winning strategy in a certain topological game, the following properties are be shown to be equivalent: (a) for every continuous f: Z \to (E,weak), where Z is an \alpha-favorable (slightly stronger than Baire) space, there exists a dense G_\delta-subset A of Z at the points of which f is norm continuous; (b) for every qc f: Z \to (E,weak), where Z is a complete metric space, there exists a point at which f is weakly continuous; (c) for every qc f: Z \to (E,weak), where Z is \alpha-favorable, there exists a dense G_\delta subset A of Z at the points of which f is norm continuous. Similar properties hold for spaces of the type C(T), with the topology of pointwise convergence on T. Some results concerning joint continuity follow.