arXiv subjects

S. Romaña

Publications and source records attributed to S. Romaña.

3 recordsLinked to original sources

arXiv2026

Topological Expansivity and Shadowing for Anosov Diffeomorphisms

We show that every Anosov diffeomorphism is topologically expansive in the sense of [9]. We also prove that Lipschitz shadowing forces the stable and unstable bundles to be uniformly transverse, with a quantitative bound in terms of the shadowing constant. Finally, we give two examples on complete Riemannian manifolds: one is expansive but does not have the shadowing property, while the other has finite volume, bounded sectional curvature, Lipschitz shadowing, and orthogonal invariant bundles, but is not expansive with respect to its Riemannian distance.

math.DS↗

arXiv2026

Affine Anosov Maps on $\mathbb{R}^n$: Classification, Index Spectrum, and Stability at Infinity

For $n\ge2$, we classify the affine diffeomorphisms $f_{A,v}(x)=Ax+v$ on $\mathbb R^n$ that admit a complete Riemannian metric with respect to which they are Anosov. Such a metric exists if and only if $A$ is hyperbolic or $f_{A,v}$ has no fixed point, equivalently $v\notin\operatorname{Im}(I-A)$. In the latter case, $f_{A,v}$ is smoothly conjugate to a translation when $\det A>0$ and to White's map times the identity when $\det A<0$. We also determine the possible stable indices. In the hyperbolic case, the index is determined by the stable spectrum of $A$, whereas a map with no fixed point admits complete Anosov metrics of every stable index from $1$ to $n-1$. Along the $1$-eigenspace, every such metric must exhibit exponential growth of vector norms along one of the two half orbits. We then determine the interior and boundary of the Anosov-realizable locus in the affine parameter space and describe the corresponding change of the index spectrum near regular drift parameters. Finally, we show that Anosov-realizability is not open in the two-sided weak $C^1_{\mathrm{loc}}$ topology but is open in the two-sided strong Whitney $C^1$ topology.

math.DS↗

arXiv2017

Continuity of Hausdorff Dimension Across Generic Dynamical Lagrange and Markov Spectra II

Let $g_0$ be a smooth pinched negatively curved Riemannian metric on a complete surface $N$, and let $Λ_0$ be a basic hyperbolic set of the geodesic flow of $g_0$ with Hausdorff dimension strictly smaller than two. Given a small smooth perturbation $g$ of $g_0$ and a smooth real-valued function $f$ on the unit tangent bundle to $N$ with respect to $g$, let $L_{g,Λ,f}$, resp. $M_{g,Λ,f}$ be the Lagrange, resp. Markov spectrum of asymptotic highest, resp. highest values of $f$ along the geodesics in the hyperbolic continuation $Λ$ of $Λ_0$. We prove that, for generic choices of $g$ and $f$, the Hausdorff dimension of the sets $L_{g,Λ, f}\cap (-\infty, t)$ vary continuously with $t\in\mathbb{R}$ and, moreover, $M_{g,Λ, f}\cap (-\infty, t)$ has the same Hausdorff dimension of $L_{g,Λ, f}\cap (-\infty, t)$ for all $t\in\mathbb{R}$.

math.DS↗