arXiv subjects

Andrea Ulliana

Publications and source records attributed to Andrea Ulliana.

3 recordsLinked to original sources

arXiv2026

Two Instances of Chaos in Deterministic and Quantum Dynamical Systems

This thesis consists of two distinct projects situated in the areas of smooth dynamics and spectral theory, respectively. They are united by a common interest in mechanisms of chaos and statistical behavior in classical and quantum dynamical systems. The first concerns smooth dynamics. We prove that all ergodic linear automorphisms of the N-dimensional torus with two-dimensional center are stably ergodic, including all ergodic automorphisms in dimensions $N \leq 5$ and $N = 7$ . This generalizes a previous result of Rodriguez-Hertz, which required an additional algebraic condition on the characteristic polynomial of the linear automorphism. The second project deals with spectral theory of Schrödinger operators. We prove that delocalization of most eigenvectors is topologically common in the space of deterministic Schrödinger Operators on a given large finite graph, provided that the IDS satisfies a suitable regularity condition. This result generalizes a recent theorem of Avila and Damanik. We also describe a flexible family of graphs satisfying our criterion, by proving a variant of the Thouless formula.

math.DS↗

arXiv2026

Geometric Properties of Higher Dimensional Solenoidal Attractors

We study the skew product systems $T: \mathbb{S}^1\times\mathbb{R}^d \to \mathbb{S}^1 \times\mathbb{R}^d$, \begin{equation*} T(x,y)=(\ell x, A y+ϕ(x)), \end{equation*} where $\ell\geq 2,$ $A \in GL_d(\mathbb{R})$ with $ρ(A)<1$, and $ϕ\in C^r(\mathbb{S}^1, \mathbb{R}^d)$. We allow the fibers to have any dimension and $A$ to be non-conformal. We show that: when $|\det(A)|\ell<1$, for almost every $ϕ$, the Hausdorff dimension of the solenoid attractor and the SRB measure equals the affinity dimension; when $|\det(A)|\ell>1$, for almost every $ϕ$, the SRB measure is absolutely continuous. We introduce a derivative dispersion condition, replacing the usual transversality condition.

math.DS↗

arXiv2026

Advances on Stable Ergodicity of Toral Automorphisms

We prove that all ergodic automorphisms of the $N$-dimensional torus with two dimensional center are stably ergodic. This includes all ergodic automorphisms in dimension $N\leq 5$ or $N=7$. This generalizes a previous result of Rodriguez-Hertz, that required an additional algebraic condition on the carachteristic polynomial of the linear automorphism. The core of the proof is a minimality criterion.

math.DS↗