Schedule for: 22w5134 - Interactions between Descriptive Set Theory and Smooth Dynamics

A buffet dinner is served daily between 5:30pm and 7:30pm in Kinnear Center 105, main floor of the Kinnear Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

A brief introduction to BIRS with important logistical information, technology instruction, and opportunity for participants to ask questions.

This is an introductory talk motivating the topics of the meeting.

This intends to be a basic course in Dynamical Systems that includes the basic terminology used in dynamics, Poincare theory on the circle, Poincare linearization and Smale's basic theory for Axiom A systems. Probably starting with expanding maps on the circle and then moving to Markov partitions for Axiom A. If possible then discuss classification of Anosov diffeos on nilmanifolds.

These talks give an introduction to Borel and analytic subsets of Polish spaces. We will explore similarities and differences between these two classes and describe structural properties of sets in them.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

Meet in the PDC front desk for a guided tour of The Banff Centre campus.

These talks give an introduction to Borel and analytic subsets of Polish spaces. We will explore similarities and differences between these two classes and describe structural properties of sets in them.

This intends to be a basic course in Dynamical Systems that includes the basic terminology used in dynamics, Poincare theory on the circle, Poincare linearization and Smale's basic theory for Axiom A systems. Probably starting with expanding maps on the circle and then moving to Markov partitions for Axiom A. If possible then discuss classification of Anosov diffeos on nilmanifolds.

These talks give an introduction to Borel and analytic subsets of Polish spaces. We will explore similarities and differences between these two classes and describe structural properties of sets in them.

A buffet dinner is served daily between 5:30pm and 7:30pm in Kinnear Center 105, main floor of the Kinnear Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

This intends to be a basic course in Dynamical Systems that includes the basic terminology used in dynamics, Poincare theory on the circle, Poincare linearization and Smale's basic theory for Axiom A systems. Probably starting with expanding maps on the circle and then moving to Markov partitions for Axiom A. If possible then discuss classification of Anosov diffeos on nilmanifolds.

We establish two new results on the Polish group of absolutely continuous homeomorphisms on the interval. Firstly, we show that there is a comeagre set of pairs of elements $\left( f,g \right)$ for which the generated subgroup $\left\langle f,g \right\rangle$ is dense. Secondly, we show that the group admits generic elements

These talks give an introduction to Borel and analytic subsets of Polish spaces. We will explore similarities and differences between these two classes and describe structural properties of sets in them.

In work with Gao, Krone, and Seward, it was shown that there does not exist a continuous lining on the free part of the shift action of $Z^n$, although there is a Borel such lining. The non-existence of a continuous lining involves an argument using a hyperaperiodic element of the space. A hyperaperiodicity argument does seem sufficient, however, to rule out k-linings for $k>1$. By combining a hyperaperiodicity argument with a forcing argument we can rule out continuous k-linings for $k>1$. However, the question for continuous $ (TCPL 201)

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

There are open sets in the space of dynamical systems free of structurally stable maps. Even in the case of surface diffeomorphisms such open sets exist (as was shown by S.Newhouse) and exhibit extremely complicated structure. We will mention several results that demonstrate some surprising features of Newhouse domains and motivate a statement that no reasonable classification is possible. Then we will formulate and go over the proofs of several results (joint with M.Foreman) that claim that a classification of all smooth dynamical systems is indeed impossible.

Meet in foyer of TCPL to participate in the BIRS group photo. The photograph will be taken outdoors, so dress appropriately for the weather. Please don't be late, or you might not be in the official group photo!

A buffet dinner is served daily between 5:30pm and 7:30pm in Kinnear Center 105, main floor of the Kinnear Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

Computer-simulated Julia sets are some of the best known mathematical images, so it comes as a surprise that some Julia sets are not computable. The interplay of the computational complexity theory and complex dynamics in the study of Julia sets leads to some beautiful mathematics. I am going to give a brief survey of the known results on computability of Julia sets, beginning with my work with M. Braverman, and ending with the most recent advances and some open problems.

Virtual participants- join us for a group photo by turning your cameras on in zoom

There are open sets in the space of dynamical systems free of structurally stable maps. Even in the case of surface diffeomorphisms such open sets exist (as was shown by S.Newhouse) and exhibit extremely complicated structure. We will mention several results that demonstrate some surprising features of Newhouse domains and motivate a statement that no reasonable classification is possible. Then we will formulate and go over the proofs of several results (joint with M.Foreman) that claim that a classification of all smooth dynamical systems is indeed impossible.

In this talk, I will present a joint work with Kostya Slutsky where we attach to any pmp (probability measure-preserving) flow a Polish group called its $L^1$ full group. These groups are invariants of $L^1$ orbit equivalence of pmp flows. To our knowledge, this notion is much less understood than its discrete version, where Belinskaya's theorem yields that $L^1$ orbit equivalence of measure-preserving transformation is equivalent to flip-conjugacy. I will compare the properties of $L^1$ full groups of pmp flows with those of $L^1$ full groups of pmp transformations and with those of orbit full groups of pmp flows.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

A buffet dinner is served daily between 5:30pm and 7:30pm in Kinnear Center 105, main floor of the Kinnear Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

I will begin by reviewing the definition and basic properties of Kakutani equivalence. Then I will give an introduction to the anti-classification techniques for isomorphism used by Foreman-Rudolph-Weiss and the generalization to Kakutani equivalence obtained by Kunde and myself.

An important question in ergodic theory dating back to the foundational paper of von Neumann is the so-called smooth realization problem: Are there smooth versions to the objects and concepts of abstract ergodic theory? Does every ergodic measure-preserving transformation have a smooth model? By a smooth model one means a smooth diffeomorphism of a compact manifold preserving a measure equivalent to the volume element which is isomorphic to the measure-preserving transformation. One of the most powerful tools of constructing smooth volume-preserving diffeomorphisms with prescribed ergodic or topological properties is the Approximation by Conjugation method introduced by Dmitri Anosov and Anatole Katok. We give an overview of the method and its contributions to the smooth realization problem. We emphasize applications of the AbC method in the context of anti-classification results. For instance, it is a key ingredient in transferring the anti-classification results of ergodic measure-preserving transformations with respect to isomorphism and Kakutani equivalence to the setting of smooth and real-analytic diffeomorphisms of the 2-torus. We also discuss a recent application showing that there is no complete numerical invariant for smooth conjugacy of orientation-preserving circle diffeomorphisms with a Liouville rotation number.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

This talk will give an overview of the theory of Polish groups, discussing their structure and actions. This will include examples (and non-examples) of universal Polish groups, the Mackey Hjorth extension theorem and universal G-spaces. We will then pass to the theory of Borel complexity and discuss the stru-ture of orbit equivalence relations induced by Polish group actions, including the existence of a maximal (or complete) such equivalence relation. Finally, we will discuss the complexity of affine homeomorphism of Choquet simplices and show its connections to several other classification problems, including some arising in topological dynamics.

This is an opportunity for people to ask open problems and in live discussion clarify points in the expository lectures and suggest methods of attacking the open problems.

A buffet dinner is served daily between 5:30pm and 7:30pm in Kinnear Center 105, main floor of the Kinnear Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

This time is saved for discussion of ideas and progress made during earlier in the week. There are three rooms available for discussion, so smaller groups can gather.

5-day workshop participants are welcome to use BIRS facilities (TCPL ) until 3 pm on Friday, although participants are still required to checkout of the guest rooms by 11AM.

This time is saved for discussion of ideas and progress made during earlier in the week. There are three rooms available for discussion, so smaller groups can gather.