## Journals

- Advances in Mathematics of Communications
- Big Data & Information Analytics
- Communications on Pure & Applied Analysis
- Discrete & Continuous Dynamical Systems - A
- Discrete & Continuous Dynamical Systems - B
- Discrete & Continuous Dynamical Systems - S
- Evolution Equations & Control Theory
- Inverse Problems & Imaging
- Journal of Computational Dynamics
- Journal of Dynamics & Games
- Journal of Geometric Mechanics
- Journal of Industrial & Management Optimization
- Journal of Modern Dynamics
- Kinetic & Related Models
- Mathematical Biosciences & Engineering
- Mathematical Control & Related Fields
- Mathematical Foundations of Computing
- Networks & Heterogeneous Media
- Numerical Algebra, Control & Optimization
- Electronic Research Announcements
- Conference Publications
- AIMS Mathematics

CPAA

We study local and global well-posedness of the initial value
problem for the Schrödinger-Debye equation in the

*periodic case*. More precisely, we prove local well-posedness for the periodic Schrödinger-Debye equation with subcritical nonlinearity in arbitrary dimensions. Moreover, we derive a new*a priori*estimate for the $H^1$ norm of solutions of the periodic Schrödinger-Debye equation. A novel phenomenon obtained as a by-product of this*a priori*estimate is the global well-posedness of the periodic Schrödinger-Debye equation in dimensions $1$ and $2$*without*any smallness hypothesis of the $H^1$ norm of the initial data in the "focusing" case.
JMD

This text is an expanded version of the lecture notes of a minicourse (with the same title of this text) delivered by the authors in the Będlewo school ``Modern Dynamics and its Interaction with Analysis, Geometry and Number Theory'' (from 4 to 16 July, 2011).

In the first part of this text, i.e., from Sections 1 to 5, we discuss the Teichmüller and moduli space of translation surfaces, the Teichmüller flow and the $SL(2,\mathbb{R})$-action on these moduli spaces and the Kontsevich--Zorich cocycle over the Teichmüller geodesic flow. We sketch two applications of the ergodic properties of the Teichmüller flow and Kontsevich--Zorich cocycle, with respect to Masur--Veech measures, to the unique ergodicity, deviation of ergodic averages and weak mixing properties of typical interval exchange transformations and translation flows. These applications are based on the fundamental fact that the Teichmüller flow and the Kontsevich--Zorich cocycle work as

In the second part, i.e., from Sections 6 to 9, we start by pointing out that it is interesting to study the ergodic properties of the Kontsevich--Zorich cocycle with respect to invariant measures other than the Masur--Veech ones, in view of potential applications to the investigation of billiards in rational polygons (for instance). We then study some examples of measures for which the ergodic properties of the Kontsevich--Zorich cocycle are very different from the case of Masur--Veech measures. Finally, we end these notes by constructing some examples of closed $SL(2,\mathbb{R})$-orbits such that the restriction of the Teichmüller flow to them has arbitrary small rate of exponential mixing, or, equivalently, the naturally associated unitary $SL(2,\mathbb{R})$-representation has arbitrarily small spectral gap (and in particular it has complementary series).

In the first part of this text, i.e., from Sections 1 to 5, we discuss the Teichmüller and moduli space of translation surfaces, the Teichmüller flow and the $SL(2,\mathbb{R})$-action on these moduli spaces and the Kontsevich--Zorich cocycle over the Teichmüller geodesic flow. We sketch two applications of the ergodic properties of the Teichmüller flow and Kontsevich--Zorich cocycle, with respect to Masur--Veech measures, to the unique ergodicity, deviation of ergodic averages and weak mixing properties of typical interval exchange transformations and translation flows. These applications are based on the fundamental fact that the Teichmüller flow and the Kontsevich--Zorich cocycle work as

*renormalization dynamics*for interval exchange transformations and translation flows.In the second part, i.e., from Sections 6 to 9, we start by pointing out that it is interesting to study the ergodic properties of the Kontsevich--Zorich cocycle with respect to invariant measures other than the Masur--Veech ones, in view of potential applications to the investigation of billiards in rational polygons (for instance). We then study some examples of measures for which the ergodic properties of the Kontsevich--Zorich cocycle are very different from the case of Masur--Veech measures. Finally, we end these notes by constructing some examples of closed $SL(2,\mathbb{R})$-orbits such that the restriction of the Teichmüller flow to them has arbitrary small rate of exponential mixing, or, equivalently, the naturally associated unitary $SL(2,\mathbb{R})$-representation has arbitrarily small spectral gap (and in particular it has complementary series).

JMD

We compute explicitly the action of the group of affine diffeomorphisms on the
relative homology of two remarkable origamis discovered respectively by Forni
(in genus $3$) and Forni and Matheus (in genus $4$). We show that, in both
cases, the action on the nontrivial part of the homology is through finite
groups. In particular, the action on some $4$-dimensional invariant subspace of
the homology leaves invariant a root system of $D_4$ type. This provides as a
by-product a new proof of (slightly stronger versions of) the results of Forni
and Matheus: the nontrivial Lyapunov exponents of the Kontsevich-Zorich
cocycle for the Teichmüller disks of these two origamis are equal to zero.

DCDS

We show that the Hausdorff dimension of stable sets of non-uniformly hyperbolic horseshoes is strictly smaller than two.

JMD

A cyclic cover of the complex projective line branched at four
appropriate points has a natural structure of a square-tiled surface.
We describe the combinatorics of such a square-tiled surface, the
geometry of the corresponding Teichmüller curve, and compute the
Lyapunov exponents of the determinant bundle over the Teichmüller
curve with respect to the geodesic flow. This paper includes a new
example (announced by G. Forni and C. Matheus in [17]
of a Teichmüller curve of a square-tiled cyclic cover in a stratum
of Abelian differentials in genus four with a maximally degenerate
Kontsevich--Zorich spectrum (the only known example in genus three
found previously by Forni also corresponds to a square-tiled cyclic
cover [15].
We present several new examples of Teichmüller curves in
strata of holomorphic and meromorphic quadratic differentials with
a maximally degenerate Kontsevich--Zorich spectrum.
Presumably, these examples cover all possible Teichmüller curves
with maximally degenerate spectra. We prove that this is indeed the case
within the class of square-tiled cyclic covers.

CPAA

The objective of this paper is two-fold: firstly, we develop a local
and global (in time) well-posedness theory for a system describing the motion
of two fluids with different densities under capillary-gravity waves in a deep
water flow (namely, a Schrödinger-Benjamin-Ono system) for

*low-regularity*initial data in both periodic and continuous cases; secondly, a family of new periodic traveling waves for the Schrödinger-Benjamin-Ono system is given: by fixing a minimal period we obtain, via the implicit function theorem, a smooth branch of periodic solutions bifurcating a Jacobian elliptic function called*dnoidal*, and, moreover, we prove that all these periodic traveling waves are nonlinearly stable by perturbations with the same wavelength.## Year of publication

## Related Authors

## Related Keywords

[Back to Top]