responsible: Alessandro Codenotti and Aleksandra Kwiatkowska
time and place: Mondays at 10:15, place Einsteinstr. 62 - M A 114 (SR 1D)
The first seminar will be on 3.04.
A continuum is a compact connected (metric) space. After proving classical results such as the boundary bumping theorem, the existence of non cut-points theorem and the Hahn-Mazurkiewicz theorem, we will focus on constructions and properties of one-dimensional continua such as dendrites, the Menger curve and the pseudo-arc. For the latter two we will prove a characterization theorem and their homogeneity. Later in the semester we will discuss selected topics on dynamics, especially entropy, of homeomorphisms of those continua. We will prove that if a chainable continuum admits a homeomorphism with positive entropy, then it must contain an indecomposable subcontinuum. Only basic knowledge of general topology is assumed.
((3.04) Introduction (simple examples of continua, indecomposable continua, intersections and inverse limits of continua, solenoids,
hyperspaces; mainly 1.8, 2.4, 2.7, 4.13 of [N]). (speaker: Aleksandra Kwiatkowska)
notes (taken by Alessandro), supplementary notes, Sierpiński carpet
(17.04) Cut-wire and boundary bumping (parts of Section 5 in [N]). (speaker: Dominik Schilke). notes
(24.04) Cut points (parts of Section 6 in [N], characterisation of [0,1] as the only subcontinuum
with exactly two noncut points). (speaker: Judit Jansat) notes
(8.05) Peano continua (speaker: Susanne Johne) notes
(15.05) Peano continua: Hahn-Mazurkiewicz Theorem (speaker: Anupam Datta) notes v1 notes v2
(22.05) Menger universal curve (Sections 2-4 [MOT]). (speaker: Andrea Vaccaro) notes
(25.05, 12-14, room SRZ 202) Menger universal curve (Sections 2-4 [MOT]).
(speakers: Andrea Vaccaro and Alessandro Codenotti)
notes (Andrea's part) notes (Alessandro's part)
(5.06) Menger universal curve (Sections 2-4 [MOT]). (speaker: Alessandro Codenotti) notes
(12.06) Dendrites (basic properties, Ważewski dendrites, entropy of homeomorphisms) (speaker: Shujie Yang)
(19.06) Dendrites (fixed point property, [N]). (speaker: Bojana Pantic) notes
(26.06) pseudo-arc [OT]. (speaker: Rob Sullivan) notes
(3.07) pseudo-arc [OT]. (speaker: Max Strohmeier) notes
(10.07) Positive entropy of homeomorphisms [M].
[N] S. Nadler, Continuum theory: an introduction, CRC Press, 1992.
[MOT] J. C. Mayer, Lex G. Oversteegen, E. D. Tymchatyn, The Menger curve Characterization and extension of homeomorphisms of non-locally-separating closed subsets, Instytut Matematyczny Polskiej Akademi Nauk, 1986.
[OT] Lex G. Oversteegen, E. D. Tymchatyn, On hereditarily indecomposable compacta, In: Banach Center Publications 1.18 (1986), pp. 407-417.
[M] C. Mouron, Positive entropy homeomorphisms of chainable continua and indecomposable subcontinua, In: Proceedings of the American Mathematical Society 139.8 (2011), pp. 2783-2791.
[DK] U. B. Darji,H. Kato, Chaos and indecomposability. Adv. Math. 304 (2017), 793-808.