Abstract: The rel foliation is a foliation of the moduli space of abelian differentials obtained by "moving the zeroes of the one form while keeping all absolute periods fixed". It has been studied in complex analysis and dynamics under different names (isoperiodic foliation, Schiffer variation, kernel foliation). Until recent years the question of its ergodicity was wide open. Recently partial results were obtained by Calsamiglia-Deroin-Francaviglia and by Hamenstadt. In our work we completely resolve the ergodicity question.
Abstract: In this talk, we study the Ahlfors regularity of planar self-affine sets under natural conditions: strong separation condition, strong irreducibility and proximality. Not surprisingly, if the dimension is strictly larger than 1, the set is never Ahlfors regular. In case if the dimension is less than or equal to 1 under the extra condition of dominated splitting, we show that the Ahfors regularity is equivalent to the positive proper dimensional Hausdorff measure and to positive proper dimensional Hausdorff measure of the projections in every Furstenberg direction.
Abstract: Horospherical flows in homogeneous spaces have been studied intensively over the last several decades and have many surprising applications in various fields. Many basic results are under the assumption that the volume of the space is finite, which is crucial as many basic ergodic theorems fail in the setting of an infinite measure space.In the talk we will discuss the infinite volume setting, and specifically, when can we expect horospherical orbits to equidistribute.
Abstract. The class (IDPFT) of nonsingular infinite direct products T of nonsingular transformations T_n, n∈N, admitting equivalent invariant probabilities, is studied. If T_n is mildly mixing for all n, the sequence of the Radon-Nikodym derivatives of T_n is asymptotically translation quasi-invariant and T is conservative then the Maharam extension of T is sharply weak mixing.
Abstract: Diophantine approximation quantifies the density of the rational numbers in the real line. The extension of this theory to algebraic numbers raises many natural questions. I will focus on a dynamical resolution to Davenport's problem and show that there are uncountably many badly approximable pairs on the parabola. The proof uses the Kleinbock--Margulis uniform estimate for nondivergence of nondegenerate curves in the space of lattices and a variant of Schmidt's game.
Abstract: We show that some examples of type-III:1 Bernoulli shifts on two symbols have a factor that is equivalent to an independent and identically distributed system and prove that there are type-III:1 Bernoulli shifts of every possible ergodic index. The latter implies that the classification of type III Bernoulli shifts according to metric isomorphism is more subtle than its classical counterpart (Ornstein theory).
Abstract: In this talk we will explore the connection between the two seemingly unrelated concepts appearing in the title. Phase transitions occur when a system undergoes an abrupt change in behaviour as a consequence of a small change in parameters. While phase transitions are evidently observed in the physical world (e.g., water freezing or evaporating), they are also ubiquitous in mathematical problems studied in statistical mechanics, probability, combinatorics and computer science.