Tridiagonal substitution Hamiltonians

We consider a family of discrete Jacobi operators on the one-dimensional integer lattice with Laplacian and potential terms modulated by a primitive invertible two-letter substitution. We investigate the spectrum and the spectral type, the fractal structure and fractal dimensions of the spectrum, exact dimensionality of the integrated density of states, and the gap structure. We present a review of previous results, some applications, and open problems. Our investigation is based largely on the dynamics of trace maps. This work is an extension of similar results on Schroedinger operators, although some of the results that we obtain differ qualitatively and quantitatively from those for the Schoedinger operators. The nontrivialities of this extension lie in the dynamics of the associated trace map as one attempts to extend the trace map formalism from the Schroedinger cocycle to the Jacobi one. In fact, the Jacobi operators considered here are, in a sense, a test item, as many other models can be attacked via the same techniques, and we present an extensive discussion on this.Comment: 41 pages, 5 figures, 81 reference

Rothberger gaps in fragmented ideals

The~\emph{Rothberger number} $\mathfrak{b} (\mathcal{I})$ of a definable ideal $\mathcal{I}$ on $\omega$ is the least cardinal $\kappa$ such that there exists a Rothberger gap of type $(\omega,\kappa)$ in the quotient algebra $\mathcal{P} (\omega) / \mathcal{I}$. We investigate $\mathfrak{b} (\mathcal{I})$ for a subclass of the $F_\sigma$ ideals, the fragmented ideals, and prove that for some of these ideals, like the linear growth ideal, the Rothberger number is $\aleph_1$ while for others, like the polynomial growth ideal, it is above the additivity of measure. We also show that it is consistent that there are infinitely many (even continuum many) different Rothberger numbers associated with fragmented ideals.Comment: 28 page

On stochastic sea of the standard map

Consider a generic one-parameter unfolding of a homoclinic tangency of an area preserving surface diffeomorphism. We show that for many parameters (residual subset in an open set approaching the critical value) the corresponding diffeomorphism has a transitive invariant set $\Omega$ of full Hausdorff dimension. The set $\Omega$ is a topological limit of hyperbolic sets and is accumulated by elliptic islands. As an application we prove that stochastic sea of the standard map has full Hausdorff dimension for sufficiently large topologically generic parameters.Comment: 36 pages, 5 figure