It is shown that for non-hyperbolic real quadratic polynomials topological and qua-sisymmetric conjugacy classes are the same. By quasiconformal rigidity, each class has only one representative in the quadratic family, which proves that hyperbolic maps are dense. Statement of the results. Dense Hyperbolicity Theorem In the real quadratic family f a (x) = ax(1 − x) , 0 < a ≤ 4 the mapping f a ha...

On one side, from the properties of Floer cohomology, invariant associated to a symplectic manifold, we define and study a notion of symplectic hyperbolicity and a symplectic capacity measuring it. On the other side, the usual notions of complex hyperbolicity can be straightforwardly generalized to the case of almost-complex manifolds by using pseudo-holomorphic curves. That’s why we study the ...

Weprovide a generalmechanism for obtaining uniform information from pointwise data. A sample result is that if a diffeomorphism of a compact Riemannian manifold has pointwise expanding and contracting continuous invariant cone families, then the diffeomorphism is an Anosov diffeomorphism, i.e., the entire manifold is uniformly hyperbolic.

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. Consider a polynomial p on R of degree d (the maximum of the degrees of the monomials in the expansion of p). We call p homogeneous if p(tw) = tp(w) for all real t and vectors w ∈...