Page 384 - 8th European Congress of Mathematics ∙ 20-26 June 2021 ∙ Portorož, Slovenia ∙ Book of Abstracts
P. 384
TOPOLOGICAL METHODS IN DYNAMICAL SYSTEMS (MS-65)
Cˇ ech cohomology and the region of influence of non-saddle sets
Héctor Barge, h.barge@upm.es
Universidad Politécnica de Madrid, Spain
This talk is devoted to show some properties of a class of isolated invariant compacta known as
non-saddle sets. We shall see that while the local dynamics near these sets is fairly easy, flows
having this kind of invariant sets may exhibit a great deal of global complexity. As we will see,
this global complexity is closely related to the way in which the non-saddle set sits in the phase
space at the cohomological level.
Densely branching trees as models for Hénon-like and Lozi-like attractors
Jan Boron´ski, boronski@agh.edu.pl
AGH University of Science and Technology, Poland
Coauthor: Sonja Štimac
Inspired by a recent work of Crovisier and Pujals on mildly dissipative diffeomorphisms of the
plane, we show that Hénon-like and Lozi-like maps on their strange attractors are conjugate
to natural extensions (a.k.a. shift homeomorphisms on inverse limits) of maps on metric trees
with dense set of branch points. In consequence, these trees very well approximate the topology
of the attractors, and the maps on them give good models of the dynamics. To the best of our
knowledge, these are the first examples of canonical two-parameter families of attractors in the
plane for which one is guaranteed such a 1-dimensional locally connected model tying together
topology and dynamics of these attractors. For Hénon maps this applies to Benedicks-Carleson
positive Lebesgue measure parameter set, and sheds more light onto the result of Barge from
1987, who showed that there exist parameter values for which Hénon maps on their attractors
are not natural extensions of any maps on branched 1-manifolds. For Lozi maps the result
applies to an open set of parameters given by Misiurewicz in 1980. Our result, that extends
to certain attractors that are homoclinic classes for mildly dissipative diffeomorphisms, can be
seen as a generalization to the non-uniformly hyperbolic world of a classical result of Williams
from 1967.
Pseudo-arc in measurable Dynamical Systems
Jernej Cˇ incˇ, cincj9@univie.ac.at
University of Vienna and IT4Innovations Ostrava, Austria
Coauthor: Piotr Oprocha
Pseudo-arc is besides the arc the only planar continuum (i.e. compact connected metric space)
so that every of its proper subcontinua is homeomorphic to itself. Its first description appeared
in the literature about hundred years ago and due to many of its remarkable properties it is an
object of much interest in several branches of mathematics. There are results indicating that
pseudo-arc appears as a generic continuum in very general settings. For instance, Bing has
proven that in any manifold M of dimension at least 2, the set of subcontinua homeomorphic
to the pseudo-arc is a dense residual subset of the set of all subcontinua of M (equipped with
382
Cˇ ech cohomology and the region of influence of non-saddle sets
Héctor Barge, h.barge@upm.es
Universidad Politécnica de Madrid, Spain
This talk is devoted to show some properties of a class of isolated invariant compacta known as
non-saddle sets. We shall see that while the local dynamics near these sets is fairly easy, flows
having this kind of invariant sets may exhibit a great deal of global complexity. As we will see,
this global complexity is closely related to the way in which the non-saddle set sits in the phase
space at the cohomological level.
Densely branching trees as models for Hénon-like and Lozi-like attractors
Jan Boron´ski, boronski@agh.edu.pl
AGH University of Science and Technology, Poland
Coauthor: Sonja Štimac
Inspired by a recent work of Crovisier and Pujals on mildly dissipative diffeomorphisms of the
plane, we show that Hénon-like and Lozi-like maps on their strange attractors are conjugate
to natural extensions (a.k.a. shift homeomorphisms on inverse limits) of maps on metric trees
with dense set of branch points. In consequence, these trees very well approximate the topology
of the attractors, and the maps on them give good models of the dynamics. To the best of our
knowledge, these are the first examples of canonical two-parameter families of attractors in the
plane for which one is guaranteed such a 1-dimensional locally connected model tying together
topology and dynamics of these attractors. For Hénon maps this applies to Benedicks-Carleson
positive Lebesgue measure parameter set, and sheds more light onto the result of Barge from
1987, who showed that there exist parameter values for which Hénon maps on their attractors
are not natural extensions of any maps on branched 1-manifolds. For Lozi maps the result
applies to an open set of parameters given by Misiurewicz in 1980. Our result, that extends
to certain attractors that are homoclinic classes for mildly dissipative diffeomorphisms, can be
seen as a generalization to the non-uniformly hyperbolic world of a classical result of Williams
from 1967.
Pseudo-arc in measurable Dynamical Systems
Jernej Cˇ incˇ, cincj9@univie.ac.at
University of Vienna and IT4Innovations Ostrava, Austria
Coauthor: Piotr Oprocha
Pseudo-arc is besides the arc the only planar continuum (i.e. compact connected metric space)
so that every of its proper subcontinua is homeomorphic to itself. Its first description appeared
in the literature about hundred years ago and due to many of its remarkable properties it is an
object of much interest in several branches of mathematics. There are results indicating that
pseudo-arc appears as a generic continuum in very general settings. For instance, Bing has
proven that in any manifold M of dimension at least 2, the set of subcontinua homeomorphic
to the pseudo-arc is a dense residual subset of the set of all subcontinua of M (equipped with
382
