Title of the talks

• Domino tilings in dimension 3

Abstract: In dimension 2, a domino is a 2x1 rectangle.

Domino tilings of quadriculated regions have been extensively studied, with several deep and famous results.

The corresponding problems in dimension 3 (or higher) appear to be almost without exception much harder.

In dimension 2, it is known, for instance, that for any quadriculated disk any two tilings can be joined by a finite sequence of flips:

a flip consists in lifting two adjacent dominos and placing them back after a quarter turn rotation.

In dimension 3, flips are not sufficient to join any two tilings of a box.

Indeed, there exist a few tilings which admit no flip. Also, there exists an invariant under flips which assumes integer values, the twist.

Under suitable hypothesis, we prove that it is almost always true that two tilings with the same twist can be joined by flips.

This talk includes joint work with several collaborators, including J. Freire, C. Klivans, P. Milet and C. Tomei.

The first result in this area, in dimension 2, is due to W. Thurston.

Nicolau Saldanha - PUC-RJ

• Invariance principle for partially hyperbolic diffeomorphisms

Abstract: Ledrappier proved that the invariant measures of linear cocycles having zero Lyapunov exponents have certain extra invariance. This was generalized by Avila and Viana for smooth cocycles, in particular they proved that the invariant measures for partially hyperbolic skew products have a disintegration invariant by holonomies, this is known as "invariance principle".

This has several applications, such as obtaining genericity of non-uniformly hyperbolic systems, finding physical measures, and classifying the measures of maximal entropy.

In this presentation we will generalize the invariance principle to partially hyperbolic non-skew products (without compact center leaves) which allows us to extend several of the previous applications to more general partially hyperbolic ones.

This is a joint work with Sylvain Crovisier.

Mauricio Poletti - UFC

•  Transfer operator for piecewise expanding maps: a unified approach

Abstract: We use the method of atomic decomposition  to study the action of transfer operators associated with piecewise expanding maps. It turns out that these transfer operators are quasi-compact even when the associated potential, the dynamics and the underlying phase space have very low regularity. In particular it is often possible to obtain exponential decay of correlations, the Central Limit Theorem and almost sure invariance principle for fairly general observables, including unbounded ones. Indeed the class of observables for which we obtain such results often coincides with certain Besov spaces. Joint work with Alexander Arbieto (UFRJ-Brazil).

Daniel Smania - ICMC- USP

•  Recent results about the phase transitions

Abstract: (thermodynamics) Phase transitions, from the Dynamical Systems point of view, often mean the lack of differentiability or analyticity of the topological pressure as a function of the potential. The information about the lack of regularity of the topological pressure function helps us to describe the thermodynamic characteristics of our dynamical system. From the works of Bowen, Smale and Ruelle, in the 70s, we know that in the hyperbolic or expanding context there is no phase transition for sufficiently regular potentials. On the other hand, there is a vast literature of examples of non-hyperbolic or expanding dynamic systems in which phase transitions occur for sufficiently regular potentials. Despite this, a complete understanding of which dynamics allow for phase transitions remains out of reach. In this lecture, we intend to expose the recent contributions that we obtained regarding this problem, both from the thermodynamic and spectral points of view. Such contributions are the result of some collaborations with Paulo Varandas, Victor Carneiro and Afonso Fernandes.

Thiago Bomfim - UFBA

• Mating quadratic maps with the modular group

Abstract: Holomorphic correspondences are multi-valued maps defined by polynomial relations P(z,w)=0. We consider a specific 1-(complex) parameter family of (2:2) correspondences (every point has 2 images and 2 preimages) which we show encodes both the dynamics of a rational map and the dynamics of the modular group. We show that the connectedness locus for this family is homeomorphic to the parabolic Mandelbrot set, itself homeomorphic to the Mandelbrot set. Joint work with S. Bullett.

Luna Lomonaco - IMPA

• On equivalences of Fourier uncertainty principles and Gaussian decay rates of harmonic oscillators

Abstract: Quite recently, we make progress on a question by Vemuri on the optimal Gaussian decay of harmonic oscillators, proving the original conjecture up to an arithmetic progression of times, combining techniques used to study the free Schrödinger equation (a machinery developed in the work of Cowling, Escauriaza, Kenig, Ponce and Vega) and a lemma which relates decay on average to pointwise decay. In particular, as a consequence of such lemma,  we get a sharp equivalences (up to the endpoint) between Hardy’s, Cowling–Price’s and Morgan’s Uncertainty Principles in the sub-critical regime. Joint work with João Pedro Ramos (ETH) and Aleksei Kulikov (NTNU).

Luscas Oliveira - UFRGS

• Invariance of entropy for maps isotopic to Anosov

Abstract: We prove the topological entropy remains constant inside the class of partially hyperbolic diffeomorphisms of T^d with simple central bundle (that is, when it decomposes into one dimensional sub-bundles with controlled geometry) and such that their induced action on the first homology group H_1(T^d) is hyperbolic. In absence of the simplicity condition we construct a robustly transitive counter-example. This is a joint work with P. Carrasco, E. Pujals and C. Vásquez.

Cristina Lizana - UFBA

• Stable intersections of Cantor sets and homoclinic bifurcations in the space of automorphisms of  C^2

• Abstract: In the 70s, Newhouse proved the existence of open sets of diffeomorphisms of surfaces having persistent homoclinic tangencies. His idea was to investigate intersections between Cantor sets related to a homoclinic bifurcation. In these Newhouse regions, many sophisticated phenomena occur, such as the coexistence of infinitely many sinks and strange attractors. Buzzard proved the existence of Newhouse regions in the space of automorphisms of C^2. However, relatively few results are known about homoclinic bifurcations in this context. In this talk, we will cover some historical results and present some contributions of mine and Gugu to this topic.

Hugo Araújo - UFOP