Topological and geometric hyperbolicity criteria for polynomial automorphisms of

We prove that uniform hyperbolicity is invariant under topological conjugacy for dissipative polynomial automorphisms of $\mathbb {C}^2$ . Along the way we also show that a sufficient condition for hyperbolicity is that local stable and unstable manifolds of saddle points have uniform geometry.

Cantor spectrum for CMV and Jacobi matrices with coefficients arising from generalized skew-shifts

We consider continuous cocycles arising from CMV and Jacobi matrices. Assuming that the Verblunsky and Jacobi coefficients arise from generalized skew-shifts, we prove that uniform hyperbolicity of the associated cocycles is $C^0$ -dense. This implies that the associated CMV and Jacobi matrices have a Cantor spectrum for a generic continuous sampling map.

Diophantine Property of Matrices and Attractors of Projective Iterated Function Systems in ℝℙ1

We prove that almost every finite collection of matrices in $GL_d( \mathbb{R} )$ and $SL_d({\mathbb{R}})$ with positive entries is Diophantine. Next we restrict ourselves to the case $d=2$. A finite set of $SL_2({\mathbb{R}})$ matrices induces a (generalized) iterated function system on the projective line ${\mathbb{RP}}^1$. Assuming uniform hyperbolicity and the Diophantine property, we show that the dimension of the attractor equals the minimum of 1 and the critical exponent.

Uniform Hyperbolicity of a Scattering Map with Lorentzian Potential

We show that a two-dimensional area-preserving map with Lorentzian potential is a topological horseshoe and uniformly hyperbolic in a certain parameter region. In particular, we closely examine the so-called sector condition, which is known to be a sufficient condition leading to the uniformly hyperbolicity of the system. The map will be suitable for testing the fractal Weyl law as it is ideally chaotic yet free from any discontinuities which necessarily invokes a serious effect in quantum mechanics such as diffraction or nonclassical effects. In addition, the map satisfies a reasonable physical boundary condition at infinity, thus it can be a good model describing the ionization process of atoms and molecules.

-openness of non-uniform hyperbolic diffeomorphisms with bounded -norm

We study the $C^{1}$-topological properties of the subset of non-uniform hyperbolic diffeomorphisms in a certain class of $C^{2}$ partially hyperbolic symplectic systems which have bounded $C^{2}$ distance to the identity. In this set, we prove the stability of non-uniform hyperbolicity as a function of the diffeomorphism and the measure, and the existence of an open and dense subset of continuity points for the center Lyapunov exponents. These results are generalized to the volume-preserving context.