Santiago Dynamical Systems seminars 2019

List of seminars 2019

http://www.dynamicalsystems.cl

10th Jan, Pseudo-Anosov diffeomorphisms

Alberto Pinto

We will introduce pseudo Cr smooth structures on surfaces that will have the following property; the Pseudo-Anosov diffeomorphisms are uniformly Cr hyperbolic. We will conjecture that the Bochi-Mane theorem will extend to such pesudo C1 smooth structures recovering the duality of the result for all surfaces.

14th Jan, Piecewice chaotic maps

Alberto Pinto

14th Jan, Piecewice chaotic maps

Alberto Pinto

We will consider the class of Cr unidimensional piecewise maps with a transitive attractor. These maps can have simultaneously discontinuities, criticalities and singularities. We will show that topological chaos is equivalent to metric chaos. We recall that this result is known in several classes strictly contained in the general class that we are presenting.

11th Mar, Construccion de aplicaciones pseudo-Anosov

Hamal Hubbard (Cornell University)

Hamal Hubbard (Cornell University)

12th Mar, Construccion de aplicaciones pseudo-Anosov

Hamal Hubbard (Cornell University)

14th Mar, Diabolical entropy

Neil Dobbs (University of Geneva)

Neil Dobbs (University of Geneva)

Milnor and Thurston proved that topological entropyas a function of parameter in the quadratic family is a monotone function. Guckhenheimer showed that it is Hölder continuous. In joint work with Nicolae Mihalache, we provide precise estimates on the Hölder exponent at almost every parameter.

18th Mar, Understanding physical mixing processes via transfer operator approach

Yiwei Zhang

Industrial and chemical mixing processes of various kinds occur throughout nature and are vital in many technological applications.In the context of discrete dynamical systems, the transfer operator approach has been shown as a powerful tools from both theoretic and numerical viewpoint.

In this talk, I will use a toy model (i.e., the one dimensional stretch and fold map) as an example to provide a brief introductionon the relationships between the spectral properties of the associated transfer operator and the estimations of the optimal mixing rate of the mixing process. Moreover, I will address how the optimal mixing rate varies according to the stretch and fold map has ``cutting and shuffling'' behaviour (i.e., composing with a permutation).

If time permits, I will also talk about how to interpret this problem to the eigenvalue estimations for the Random bi-stochastic matrices (free probability theory) and the locations of poles of the dynamical zeta function.

25th Mar, Cantor dynamics and simple left-orderable groups

Michele Triestino

Michele Triestino

I will present a construction of simple groups of homeomorphisms of the real line. Given a homeomorphism of a Cantor set \sigma: X --> X, consider the suspension Y=X x [0,1]/ (x,1)~(\sigma(x),0), and look at the group H_0(Y) of homeomorphisms of Y, isotopic to the identity. If \sigma is minimal, then H_0(Y) is simple [Aliste-Prieto - Petite], and I will describe countable subgroups T(Y) which are also simple.

These are reminiscent of the classical Thompson groups, and feature several nice properties. For instance, when \sigma is a minimal subshift, T(Y) is finitely generated. Joint work with Nicolás Matte Bon.

These are reminiscent of the classical Thompson groups, and feature several nice properties. For instance, when \sigma is a minimal subshift, T(Y) is finitely generated. Joint work with Nicolás Matte Bon.

1st Apr, Counting problem on infinite periodic billiards and translation surfaces

Angel Pardo

The Gauss circle problem consists in counting the number of integer points of bounded length in the plane. This problem is equivalent to counting the number of closed geodesics of bounded length on a flat two dimensional torus or, periodic trajectories, in a square billiard table.

Many counting problems in dynamical systems have been inspired by this problem. For 30 years, the experts try to understand the asymptotic behavior of closed geodesics in translation surfaces and periodic trajectories on rational billiards. (Polygonal billiards yield translation surfaces naturally through an unfolding procedure.) H. Masur proved that this number has quadratic growth rate.

In these talk, we will study the counting problem on infinite periodic rational billiards and translation surfaces.

The first example and motivation is the wind-tree model, a Z^2-periodic billiard model. In the classical setting, we place identical rectangular obstacles in the plane at each integer point; we play billiard on the complement.

I will first present some quite precise results that are only valid for the wind-tree model (and some natural generalizations) and then, a general result which is valid for a.e. infinite periodic translation surfaces that uses completely different techniques: a dynamical analogous, for the algebraic hull of a cocycle, to strong and super-strong approximation on algebraic groups.

8th Apr,**Renormalization of multicritical circle maps**

Gabriela Estevez

We study $C^3$ orientation preserving circle homeomorphisms with irrational rotation number and non-flat critical points. By Yoccoz, two of these maps with same irrational rotation are topologically conjugate. In this talk, we define the Renormalization operator of this kind of maps and assuming some properties of this operator we prove that the conjugacy is a $C^{1+\alpha}$ diffeomorphism. This result is valid for a total Lebesgue measure set of irrational rotation numbers. This is a joint work with Pablo Guarino (Universidade Federal Fluminense, Brazil).

Angel Pardo

The Gauss circle problem consists in counting the number of integer points of bounded length in the plane. This problem is equivalent to counting the number of closed geodesics of bounded length on a flat two dimensional torus or, periodic trajectories, in a square billiard table.

Many counting problems in dynamical systems have been inspired by this problem. For 30 years, the experts try to understand the asymptotic behavior of closed geodesics in translation surfaces and periodic trajectories on rational billiards. (Polygonal billiards yield translation surfaces naturally through an unfolding procedure.) H. Masur proved that this number has quadratic growth rate.

In these talk, we will study the counting problem on infinite periodic rational billiards and translation surfaces.

The first example and motivation is the wind-tree model, a Z^2-periodic billiard model. In the classical setting, we place identical rectangular obstacles in the plane at each integer point; we play billiard on the complement.

I will first present some quite precise results that are only valid for the wind-tree model (and some natural generalizations) and then, a general result which is valid for a.e. infinite periodic translation surfaces that uses completely different techniques: a dynamical analogous, for the algebraic hull of a cocycle, to strong and super-strong approximation on algebraic groups.

8th Apr,

Gabriela Estevez

We study $C^3$ orientation preserving circle homeomorphisms with irrational rotation number and non-flat critical points. By Yoccoz, two of these maps with same irrational rotation are topologically conjugate. In this talk, we define the Renormalization operator of this kind of maps and assuming some properties of this operator we prove that the conjugacy is a $C^{1+\alpha}$ diffeomorphism. This result is valid for a total Lebesgue measure set of irrational rotation numbers. This is a joint work with Pablo Guarino (Universidade Federal Fluminense, Brazil).

15th Apr, Multifractal analysis for self-affine systems

Thomas Jordan

Joint work with Balazs Barany, Antti Kaenmaki and Michal Rams. If you consider a uniformly expanding Markov map on the interval and a continuous function. You can consider level sets of point for which the Birkhoff average is some fixed point. A typical problem I need multifractal analysis is to look at the dimension of these level sets. We will show how this can be done using the topological pressure and then how results can be obtained in the setting of certain self-affine sets in two dimensions using the sub-additive pressure and approximation by dominated subsystems.

22th Apr, **On the dynamics of elliptic functions of the form P+b**

**Monica Moreno Rocha**

The dynamical system obtained by iteration of the Weierstrass P function over real square lattices can be characterized by the behavior of its single free critical orbit. In contrast, as soon as P is “perturbed” by the addition of a complex parameter b, the elliptic function P+b exhibits at least two free critical orbits, which complicates the study of its dynamics and connectedness locus. This talk I will present some of the results and open questions regarding the rich structures found in dynamical and parameter plane of P+b when b is restricted to a complex line and P is defined over real square lattices. This is a joint work with Jane M. Hawkins, UNC-Chapel Hill.

6th May, **Algebraic invariant of minimal group actions on the Cantor set: topological full group and group of automorphism**

**María Isabel Cortez**

TBA

TBA

13th May, **Assouad dimension of planar self-affine sets**

**Antti Käenmäki**

We consider planar self-affine sets X satisfying the strong separation condition and the projection condition. We show that any two points of X, which are generic with respect to a self-affine measure having simple Lyapunov spectrum, share the same collection of tangent sets. We also calculate the Assouad dimension of X. Finally, we prove that if X is dominated, then it is minimal for the conformal Assouad dimension. The talk is based on joint work with Balázs Bárány and Eino Rossi.

27th May, TBA

Ryo Moore

TBA

3th Jun, **TBA**

TBA

10th Jun, **TBA**

**TBA**

TBA

17th Jun, **TBA**

**TBA**

TBA

24th Jun, **TBA**

**TBA**

TBA

TBA

1th Jul, **TBA**

**TBA**

TBA

8th Jul, **TBA**

TBA

15th Jul, **TBA**

**TBA**

TBA

TBA

22th Jul, **TBA**

**TBA**

TBA

29th Jul,**TBA**

**TBA**

TBA

29th Jul,

TBA

5th Aug, **TBA**

**TBA**

TBA

12th Aug,**TBA**

**TBA**

TBA

12th Aug,

TBA

19th Aug,

TBA

26th Aug,**TBA**

**TBA**

26th Aug,

TBA

2nd Sep, **TBA**

**TBA**

TBA

Dynamical Systems seminars 2019

Italo Cipriano

I am in charge of the
__seminar of Dynamical Systems in Santiago __
. Details of each seminar 2019 are available
__here__.
Seminars 2018 are available
__here__
.

I am a postdoctoral fellow supported by CONICYT PIA ACT172001 at Pontificia Universidad Católica de Chile. My research is in Dynamical Systems and I am primarily interested in Thermodynamic Formalism.

Time change for flows and thermodynamic formalism (with Godofredo Iommi). (2019) (accepted in Nonlinearity).

Stationary measures associated to analytic iterated function schemes (with Mark Pollicott). Math. Nachr. 291 (2018), no. 7, 1049–1054.

Entry time statistics to different shrinking sets, Stoch. Dyn., 17 (2017), no. 3, 314–323.

**Preprints**

Entry time statistics to different shrinking sets, Stoch. Dyn., 17 (2017), no. 3, 314–323.

Continuous coboundaries of the product of smooth functions (with Ryo Moore). (arXiv).

The Wasserstein distance between stationary measures associated to iterated function schemes on the unit interval. (arXiv).

The smoothness of the stationary measure. (arXiv).

A Large deviation and an escape rate result for special semi-flows. (arXiv).

Escape rate for special semi-flows over non-invertible subshifts of finite type. (arXiv).

Fall and Spring 2017. Differentiation and Integration, Universidad Técnica Federico Santa María.

Fall and Spring 2017. Introduction to Calculus, Universidad Técnica Federico Santa María.

Fall 2017. Foundations, Universidad Adolfo Ibáñez.

Spring 2016. Linear Algebra, Universidad Adolfo Ibáñez.