IMEUSP Dynamical Systems Seminar [pt]
Instituto de Matemática e Estatística
Universidade de São Paulo
Rua do Matão, 1010
Cidade Universitária
São Paulo, SP
Brasil  CEP 05508090
Archived Material.
Programme 2017. [en] [pt]
Programme 2018.

2018/04/10 at 16:00 (Room 132A  A Block  IMEUSP)
 Luna Lomonaco (IMEUSP)
 Title: Correspondences in Dynamics
 Abstract:
In 1994, S. Bullett and C. Penrose introduced the one complex parameter family of (2:2) holomorphic correspondences F_{a}:
and proved that for every value of a ∈ [4,7] ⊂ ℝ the correspondence F_{a} is a mating between a quadratic polynomial
Q_{c}(z)=z^{2}+c, c ∈ ℝ ,
and the modular group
PSL(2,Z).
They conjectured that this is the case for every member of the family F_{a} which has the parameter a in the connectedness locus.
We show here that matings between the modular group and rational maps in the parabolic quadratic family Per_{1}(1) provide a better model:
we prove that every member of the family F_{a} which has the parameter a in the connectedness locus is such a mating. Moreover, we develop a dynamical
theory for such a family which parallels the DouadyHubbard theory of quadratic polynomials.

2018/04/24 Cancelled

2018/04/24 at 16:00 (Room 132A  A Block  IMEUSP)
 Peter Hazard (IMEUSP)
 Title: Zeta functions, hyperbolic toral automorphisms and Lévy constants
 Abstract:
The Lévy constant of an irrational real number, when it exists, is a positive real number which measures the rate of growth of the
denominators of the sequence of best rational approximants. It has other geometric interpretations, including the rate of convergence to the
boundary of certain geodesics in the upper halfplane or the length of closed geodesics on the modular surface.
In this lecture I will show a link between Lévy constants and the topological entropy and dynamical zeta functions of certain hyperbolic
toral automorphisms, and the scenery flow of the torus.

2018/05/22 at 16:00 (Room 132A  A Block  IMEUSP)
 Daniel Meyer (Liverpool University, UK)
 Title: Topics in Complex Dynamics: Quasisymmetry, expanding Thurston maps, and matings (Lecture 1 of 3)
 Abstract:
Lecture 1 of 3 of a minicourse based on the following topics:
 Snowballs, quasisymmetric uniformization,
 Rough geometry, Gromov hyperbolic spaces, visual metrics and quasisymmetry,
 Expanding Thurston maps,
 Mating/unmating of rational maps.

2018/05/24 at 16:00 (Room 249A  A Block  IMEUSP)
 Daniel Meyer (Liverpool University, UK)
 Title: Topics in Complex Dynamics: Quasisymmetry, expanding Thurston maps, and matings (Lecture 2 of 3)
 Abstract:
Lecture 2 of 3. A continuation of the minicourse.

2018/05/29 at 16:00 (Room 132A  A Block  IMEUSP)
 Daniel Meyer (Liverpool University, UK)
 Title: Topics in Complex Dynamics: Quasisymmetry, expanding Thurston maps, and matings (Lecture 3 of 3)
 Abstract:
Lecture 3 of 3. A continuation of the minicourse.

2018/06/05 at 16:00 (Room 132A  A Block  IMEUSP)
 Zbigniew Nitecki (Tufts University, USA)
 Title: Hyperbolicity of Diffeomorphisms on Compact and Noncompact Spaces
 Abstract:
TBA.

2018/06/12 at 16:00 (Room 132A  A Block  IMEUSP)
 Jérôme Los (Université AixMarseille, France)
 Title: On BowenSerieslike Maps (Lecture 1 of 2)
 Abstract:
Lecture 1 of 2. In this series of talks I will present a construction originally due to R. Bowen and C. Series that creates a relationship between a group and a dynamical system. This relationship is potentially more general than the strict situation described here : surface groups.
I will describe a combinatorial construction with some consequences, in particular to growth properties on groups.

2018/06/14 at 16:00 (Room 249A  A Block  IMEUSP)
 Jérôme Los (Université AixMarseille, France)
 Title: On BowenSerieslike Maps (Lecture 2 of 2)
 Abstract:
Lecture 2 of 2. A continuation of Tuesday's lecture.

2018/06/19 at 16:00 (Room 132A  A Block  IMEUSP)
 Danilo Caprio (UNESPIBILCE)
 Title: Stochastic Vershik Map and filled Julia sets [CANCELLED]
 Abstract:
In this lecture, we define some Markov Chains associated to Vershik maps on Bratteli diagrams.
We study probabilistic and spectral properties of their transition operators and we prove that the spectra of these operators are connected to filled
Julia sets of polynomial maps in higher dimensions. We also study topological properties of these spectra.
This is a joint work with Ali Messaoudi and Glauco Valle.

2018/06/19 at 16:00 (Room 132A  A Block  IMEUSP)
 Jérôme Los (Université AixMarseille, France)
 Title: Sequences in MCG(2) and induction
 Abstract: TBA

2018/06/21 at 16:00 (Room 249A  A Block  IMEUSP)
 Zbigniew Nitecki (Tufts University, USA)
 Title: Crossing matrices for braids
 Abstract: TBA

2018/06/26 at 16:15 (Room 132A  A Block  IMEUSP)
 Pierre Arnoux (Institut de Mathématiques de Luminy, Marseille, France)
 Title: Multidimensional continued fractions and symbolic dynamics for toral translations
 Abstract:
We give a dynamical and geometric interpretation to multidimensional continued fractions algorithms.
For some strongly convergent algorithms, the construction gives symbolic dynamics of sublinear complexity for almost all toral translations;
it can be used to obtain a symbolic model of the diagonal flow on lattices in ℝ ^{3}, and a nonstationary Markov partition for Anosov families
on the 3torus associated with orbits of these continued fraction.

2018/07/03 at 16:00 (Room 132A  A Block  IMEUSP)
 Carsten Lunde Petersen (Roskilde University, Denmark)
 Title: On Combinatorial types of cycles under under z^{d}
 Abstract:
The talk is based on joint work with Saeed Zakeri. Rotation sets for z^{d}, sets on which z^{d}
is topologically conjugate to a rigid rotation,
are well studied in the literature. Much less is known about periodic orbits of other types of combinatorics.
To be precise by a combinatorics (of period q) we mean a dynamics on
0< x_{1} < x_{2} < ... < x_{q} < 1 ∈ T := R/Z
fixing 0 ≡ 1 and
which acts as a permutation of order q on the x_{i}.
Which combinatorics are realized under z^{d}?
In how many distinct ways is a given combinatorics realized?
How does this number depend on the degree d?
We give complete answers to these questions in simple terms.
The answers generalize works of Bullet and Sentenac for degree 2 and of McMullen.

2018/07/27 at 17:15 (Auditorium Antonio Giglioli  A Block  IMEUSP)
 Mitsuhiro Shishikura (Kyoto University, Japan)
 Title: The Hausdorff dimension of the boundary of the Mandelbrot set and parabolic points
 Abstract:
The Mandelbrot set M is a fractal set, which appears in the parameter space of complex quadratic polynomials
p_{c}(z)=z^{2}+c.
Its boundary is characterized as the set of parameters where the bifurcation occurs.
We show that the boundary of M has Hausdorff dimension two (maximal among the subset of the plane).
Key ingredients are the comparison between the phase space and the parameter space,
the analysis of the bifurcation of parabolic fixed points (whose derivative is a root of unity).
Peter Hazard
<pete at ime dot usp dot br>.
Last revised July 2018.