Мета курсу: The study of topological transitivity, minimality, chaotic
properties, topological entropy and functional envelopes for discrete
dynamical systems given by compact Hausdorff (metric) spaces and
continuous selfmaps.
Приблизний перелік тем курсу:
1. Topological transitivity and minimality:
Topologically transitive maps
Minimal maps
Minimal sets and spaces
2. Li-Yorke sensitivity and other concepts of chaos:
On chaotic interval maps
Topological chaos and Li-Yorke chaos
Li-Yorke sensitivity and weakly mixing maps
On Lyapunov numbers
3. Topological entropy:
Topological entropy of (nonautonomous) dynamical systems
Topological entropy of (nonautonomous) piecewise-monotone dynamical
systems on the interval and applications
Group homeomorphisms and topological entropy of their elements
4. Functional envelope of a dynamical system:
Introduction and topological transitivity
Topological entropy of a functional envelope
Література:
S. Kolyada Topological dynamics: minimality, entropy and chaos
(Ukrainian), Proceedings of Inst. Math., NAS of Ukraine, Mathematics
and its Applications, 89, Kiev, 2011, 340 pp.
S. Kolyada and L. Snoha, Some aspects of topological transitivity - a
survey, Proc. ECIT-94, Grazer Mathematische Berichte, 334 (1997), 3-35
S. Kolyada, L. Snoha and S. Trofimchuk, Noninvertible minimal maps,
Fundamenta Mathematicae, 168 (2001), 141--163,
F. Blanchard, E. Glasner, S. Kolyada and A. Maass, On Li-Yorke pairs,
Journal für die reine und angewandte Mathematik (Crelle's Journal),
547 (2002), 51--68,
S. Kolyada, Li-Yorke sensitivity and other concepts of chaos,
(Ukrainian) Ukrain. Mat. Zh. 56 (2004), 1043--1061; translation in
Ukrainian Math. J. 56 (2004), 1242--1257.
Передумови навчання: Володіння основами курсів теорії ймовірностей,
топології та функціонального аналізу.