Pesin Theory-An important branch of dynamical systems and of smooth ergodic theory, with many applications to non-linear dynamics. The name is due to the landmark work of Yakov B. Pesin in the mid-seventies 20, 21, 22]. Sometimes it is also referred to as the theory of smooth dynamical systems with non-uniformly hyperbolic behavior, or simply theory of non-uniformly hyperbolic dynamical systems...

By implementing jet differential techniques in non-archimedean geometry, we obtain a big Picard type extension theorem, which generalizes previous result of Cherry and Ru. As applications, establish two hyperbolicity-related results. Firstly, prove Ax-Lindemann theorem for totally degenerate abelian varieties. Secondly, show the pseudo-Borel hyperbolicity subvarieties general

Gromov hyperbolicity of a metric space measures the distance from perfect tree-like structure. The measure has “worst-case” aspect to it, in sense that it detects region which sees maximum deviation In this article we introduce an “average-case” version hyperbolicity, whether “most space”, with respect given probability measure, looks like tree. main result paper is if average small, then can b...

Pesin Theory-An important branch of dynamical systems and of smooth ergodic theory, with many applications to non-linear dynamics. The name is due to the landmark work of Yakov B. Pesin in the mid-seventies 20, 21, 22]. Sometimes it is also referred to as the theory of smooth dynamical systems with non-uniformly hyperbolic behavior, or simply theory of non-uniformly hyperbolic dynamical systems...

At the study of non-Gaussian signatures in CMB we recently have shown that the low density regions, the voids, can induce hyperbolicity properties to null geodesics in globally flat and slightly positively curved Universe with matter inhomogeneities. In terms of an introduced porosity parameter, we now obtain the criterion for hyperbolicity: high underdensity regions, i.e. almost empty voids an...

The Generalized Lax Conjecture asks whether every hyperbolicity cone is a section of a semidefinite cone of sufficiently high dimension. We prove that the space of hyperbolicity cones of hyperbolic polynomials of degree d in n variables contains (n/d) pairwise distant cones in the Hausdorff metric, and therefore that any semidefinite representation of such polynomials must have dimension at lea...

In 1958 Lax conjectured that hyperbolic polynomials in three variables are determinants of linear combinations of three symmetric matrices. This conjecture is equivalent to a recent observation of Helton and Vinnikov. A homogeneous polynomial p on R n is hyperbolic with respect to a vector e ∈ R n if p(e) = 0 and, for all vectors w ∈ R n , the univariate polynomial t → p(w − te) has all real ro...

Considering n fluids, which are homogeneous, inviscid, incompressible and without surface tension, the 3D Euler equations with long wave assumptions provide the multi-layer shallow-water model with free surface. Since salinity is assumed discontinuous, these equations can be used to describe the oceans. The study of the hyperbolicity of this quasi-linear PDEs system is divided in two parts. Fir...

Let S be a surface with genus g and n boundary components and let d(S) = 3g − 3 + n denote the number of curves in any pants decomposition of S. We employ metric properties of the graph of pants decompositions CP(S) prove that the Weil-Petersson metric on Teichmüller space Teich(S) is Gromov-hyperbolic if and only if d(S) ≤ 2. When d(S) ≥ 3 the Weil-Petersson metric has higher rank in the sense...