Wandering Seminar
Ergodic Theory & Dynamical Systems
Wroclaw University of Technology
Wroclaw, 23–26 April 2015
http://prac.im.pwr.wroc.pl/~wandering/
Abstracts of talks
April 20, 2015
Mini-course, Friday–Sunday, April 24–26
Smooth dynamics and their measures of large entropy
J´
erˆ
ome Buzzi (Universit´
e Paris-Sud)
Entropy is a classical and fundamental invariant in dynamics. For low dimensional smooth
dynamics (and other interesting classes), positive entropy yields hyperbolicity in the sense of
Pesin, and even, as recently discovered, a symbolic dynamics (Sarig, strengthened in a joint
work with Boyle). This yields ”complete classification” (Hochman): measures maximizing
the entropy (in some generalized sense) are determined by their entropy and period and in
turn determine all aperiodic invariant probability measures. After reviewing some basics of
countable state Markov shifts and hyperbolic dynamics, we will explain (some of) the proofs
of these recent results. As much as time permits, we will try and discuss some results in
smooth dynamics about these ”entropy-period” invariants.
Lectures:
1. Overwiew
2. Countable state Markov shifts: Gurevich entropy; mme; almost Borel universality
(Hochman)
3. Some basics of uniform and Pesin hyperbolicity
4. Existence of mme: smoothness; hyperbolicty; counter-examples
5. Surface diffeomorphisms: symbolic covers (Sarig) and almost Borel classification
Eventual additional topics:
- variation of the entropy: continuity properties and local constancy (geometric structures)
- partially hyperbolic diffeomorphisms with central dimension 1
Introductory lecture, Thursday, April 23
Dominik Kwietniak
This talk is intended to introduce/refresh some prerequisites for the minicourse presented
by J´erˆome Buzzi. We will cover the following topics:
• Shifts of finite type (vertex-shifts, topological entropy, mem)
• Anosov diffeomorphisms, example of coding for a two dimensional (linear) hyperbolic
toral automorphism, structural stability in this case
• Entropy
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Talks (Friday–Sunday, April 24–26) in alphabetical order
Entropies of hyperbolic groups
Andrzej Bi´
s
Hyperbolic groups in the Gromovs sense play an important role in geometric group theory.
To every hyperbolic group one can associate its boundary which has a very rich topological,
quasi-conformal and dynamical structure. A hyperbolic group acts on the boundary of its
Cayley graph. In the talk some upper and lower estimations of the topological entropy of
a hyperbolic group acting on its ideal boundary will be provided. Also, some applications
to dynamics of foliated spaces will be presented. The talk is based on the joint paper with
Pawe Walczak.
Structure theorems for Host-Kra factors of finitely generated abelian
actions
Yonatan Gutman
The factors introduced by Host & Kra (and by Ziegler using a different framework) for an
ergodic system are higher order analogues of the classical Kronecker factor. An important
fact is that they are characteristic for various non-conventional ergodic averages including the
one used by Furstenberg in order to establish Szemeredi’s theorem. A key structural result
implies that these factors are (uniquely ergodic) inverse limits of nilsystems - in particular
(just as in the Kronecker case) belong to topological category. We propose a new way
of ”passing from the measurable to the topological” which has the advantage of working
for finitely generated abelian ergodic actions. The main tools are the Camarena-Szegedy
concept of cocycle and a generalization of the Host-Kra-Maass structure theorem in a recent
joint work with Freddie Manners and Peter Varju. As an application one finds characteristic
factors for various non-conventional ergodic averages in the context of finitely generated
abelian ergodic actions.
Non-Shannon inequalities and entropy regions
Michal Kupsa
For a joining of multiple stationary processes (or measure-theoretical dynamical systems),
we consider join entropies of all the subcollection of the processes. A finite-dimensional
vector of real non-negative numbers arises in this way. The question we try to answer is how
the set of all such vectors look like. There are usual constraints on the set following from
non-negativity, monotonicity, subadditivity and conditional subadditivity of the entropy.
These constraints completely determines the set when the number of processes is at most
three. Surprisingly, another type of constraints appeared when four or more processes are
considered. These constrains are expressed via linear inequalities, called non-Shannon. Some
of inequalities of this type will be presented in this talk.
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Around the specification property
Dominik Kwietniak
A dynamical system is intrinsically ergodic if it has a unique measure of maximal entropy.
Bowen introduced the specification property and proved that for expansive systems it implies
intrinsic ergodicity. Pfister and Sullivan generalized Bowen’s specification and defined the
g-almost product property, later renamed the almost specification property by Thompson.
It was an open question whether every almost specified shift space is intrinsically ergodic. I
am going to present the example showing that the answer is negative, along with some other
results about almost specified dynamical systems. Time permits I will discuss some results
on intrinsic ergodicity of subordinate shift spaces. The talk is based on a joint work with
Piotr Oprocha (AGH Krak´ow) and MichalRams (IM PAN Warszawa).
Generic Points and the Besicovitch Pseudometric
Martha L¸
acka
Let MT (X) denote the simplex of invariant measures of the dynamical system (X, T ). For
any point x ∈ X and any positive integer n denote by m(x, n) the measure 1/n(δ(x) + . . . +
δ(T n−1 (x))) and by ω
ˆ (x) the set of all accumulation points of the sequence {m(x, n)}n∈N
with respect to the weak-* topology. During the talk we will show that if a dynamical
system has the asymptotic average shadowing property, then for every compact connected
and non-empty set V ⊂ MT (X) there exists x ∈ X such that ω
ˆ (x) is equal to V . In
particular it means that every invariant measure has a generic point. In the proof we will
use the Besicovitch pseudometric.
The talk is based on the joined work with Dominik Kwietniak and Piotr Oprocha.
On multi-recurrence and families of sets of integers
Piotr Oprocha
In 1970s Furstenberg observed that there are tight connections between recurrence in dynamics and properites of sets of integers. A classical application of this approach is Multiple
Recurrence Theorem which can be used to provide ”dynamical” proof of van der Waerden
theorem, and at he same time can be derived from this theorem (in this sense both theorems
are equivalent).
We say that a point x ∈ X is multi-recurrent if it satisfies the conclusion of the topological
multiple recurrence theorem, that is for any d ∈ N there is a strictly increasing sequence
ink
{nk }∞
x → x as k → ∞ for every i = 1, 2, . . . , d.
k=1 in N with T
In this talk we will characterize some properties of multi-recurrent points and their relations
to families of sets of integers.
Periodic orbits for real planar polynomial vector fields of degree n
having n invariant straight lines
Ana Rodrigues
In this talk, we will study the existence and non-existence of periodic orbits and limit cycles
for planar polynomial differential systems of degree n having n real invariant straight lines
taking into account their multiplicities. This is joint work with Jaume Llibre.
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Integrability and non-integrability in second order dynamical systems
on homogenous spaces
Maciej P. Wojtkowski
Lecture 1
First and second order equations on Lie groups. Euler equations. Physical example: the
rigid body dynamics.
Lecture 2
Non-standard proof of the integrability of the rigid body dynamics, and how it can be
generalized. Special automorphism of Lie groups and reversibility in dynamics.
Lecture 3
The special class of reversible second order dynamical systems on Lie groups. Integrability
and partial integrability. Presence and absence of first integrals. Positive Lyapunov exponents. Real analytic dynamical systems with invariant measures with C ∞ densities and no
real analytic density.
We gratefully acknowledge financial support from the following partner organisations:
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