Continuum Theory and Topological Dynamics

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.

Topics to cover:

((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].

References:

[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.