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Alexandro Luna

Publications and source records attributed to Alexandro Luna.

6 recordsLinked to original sources
arXiv2025

On the Regularity of the Dimension of Cookie-Cutter-Like Sets

We study the Hausdorff and box-counting dimensions of cookie-cutter-like sets formed by sequential dynamics of a finite number of expanding maps. Under some natural conditions, these dimensions turn out to be the minimum and maximum of the corresponding dimensions of the cookie-cutter sets generated by the individual expanding maps. In the case of one-parameter families of such systems, this provides a simple mechanism for producing non-differentiable fractal dimensions as functions of the parameter. This supports a conjecture that the Hausdorff dimension of the spectrum of a Sturmian Hamiltonian, in general, does not have to be differentiable as a function of the coupling constant. This is in drastic contrast to the analytic dependence of the dimensions of such spectra with quadratic irrational frequencies, e.g. the Fibonacci Hamiltonian, previously shown by M. Pollicott.

math.DS↗
arXiv2025

On the Spectra of Sieved Schr\"odinger Operators

We give a family of examples of discrete Schr\"odinger operators whose spectral dimension is not invariant under sieving. The examples are produced from the Fibonacci Hamiltonian, which is one of the main models of a one-dimensional quasicrystal. We also give a family of examples in which the local Hausdorff dimension tends to zero in some parts of the spectrum as the sieving parameter is sent to infinity.

math.SP↗
arXiv2024

Generalized Fiber Contraction Mapping Principle

We prove a generalized non-stationary version of the fiber contraction mapping theorem. It was originally used in [HirschPugh70] to prove that the stable foliation of a $C^2$ Anosov diffeomorphism of a surface is $C^1$. Our generalized principle is used in [Luna24], where an analogous regularity result for stable foliations of non-stationary systems is proved. The result is stated in a general setting so that it may be used in future dynamical results in the random and non-stationary settings, especially for graph transform arguments.

math.DS↗
arXiv2024

Regularity of Non-stationary Stable Foliations of Toral Anosov Maps

We consider a sequence of $C^2$ (or $C^3$) Anosov maps of the two-dimensional torus that satisfy a common cone condition, and show that if their $C^2$ (respectively, $C^3$) norms are uniformly bounded, then the non-stationary stable foliation must be of class $C^1$ (respectively, $C^{1+\text{H\"older}}$). This generalizes the classical results on smoothness of the invariant foliations of Anosov maps. We also provide an example that shows that an assumption on boundedness of the norms cannot be removed, which is a phenomenon that does not have an analog in the stationary setting. The main motivation stems from a standing conjecture concerning the dimension properties of the spectra of Sturmian Hamiltonian operators, and this result serves as a first step towards addressing this conjecture. A detailed appendix is provided showing the potential argument and connection between this theory of non-stationary hyperbolic dynamics and the spectral dimension of these operators. We also provide an addendum demonstrating that a similar result holds for a sequence of Anosov maps of the $d$-dimensional torus whose stable directions have codimension $1$.

math.DS↗