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Rodrigo A. Pérez

Publications and source records attributed to Rodrigo A. Pérez.

6 recordsLinked to original sources
arXiv2026

Sofic systems and extreme points of self-affine limit sets

Given any iterated function system (IFS) on the plane whose affine transformations share the same contracting matrix $T$, we characterize the set of extreme points of the attractor $A$ (along the convex hull $\operatorname{conv}(A)$) in terms of a directed graph on the edges of $\operatorname{conv}(A)$. This graph corresponds to a sofic system on the coding space associated to the IFS. When $T$ is proper orthogonal or purely real, we give conditions for $\operatorname{conv}(A)$ to be a finite polygon and, under additional symmetry of the transformations, we give conditions for $A$ itself to coincide with $\operatorname{conv}(A)$.

math.DS↗
arXiv2021

Accessibility of the Boundary of the Thurston Set

Consider two objects associated to the Iterated Function System (IFS) $\{1+λz,-1+λz\}$: the locus $\mathcal{M}$ of parameters $λ\in\mathbb{D}\setminus\{0\}$ for which the corresponding attractor is connected; and the locus $\mathcal{M}_0$ of parameters for which the related attractor contains $0$. The set $\mathcal{M}$ can also be characterized as the locus of parameters for which the attractor of the IFS $\{1+λz, λz, -1+λz\}$ contains $λ^{-1}$. Exploiting the asymptotic similarity of $\mathcal{M}$ and $\mathcal{M}_0$ with the respective associated attractors, we give sufficient conditions on $λ\in\partial\mathcal{M}$ or $\partial\mathcal{M}_0$ to guarantee it is path accessible from the complement $\mathbb{D}\setminus\mathcal{M}$.

math.DS↗
arXiv2020

Flexibility of entropies for piecewise expanding unimodal maps

We investigate the flexibility of the entropy (topological and metric) for the class of piecewise expanding unimodal maps. We show that the only restrictions for the values of the topological and metric entropies in this class are that both are positive, the topological entropy is at most $\log 2$, and by the Variational Principle, the metric entropy is not larger than the topological entropy. In order to have a better control on the metric entropy, we work mainly with topologically mixing piecewise expanding skew tent maps, for which there are only 2 different slopes. For those maps, there is an additional restriction that the topological entropy is larger than $\frac{1}{2}\log2$. We also make the interesting observation that for skew tent maps the sum of reciprocals of derivatives of all iterates of the map at the critical value is zero. It is a generalization and a different interpretation of the Milnor-Thurston formula connecting the topological entropy and the kneading determinant for unimodal maps.

math.DS↗
arXiv2003

Geometry of $Q$-recurrent maps

Given a critically periodic quadratic map with no secondary renormalizations, we introduce the notion of $Q$-recurrent quadratic polynomials. We show that the pieces of the principal nest of a $Q$-recurrent map $f_c$ converge in shape to the Julia set of $Q$. We use this fact to compute analytic invariants of the nest of $f_c$, to give a complete characterization of complex quadratic Fibonacci maps and to obtain a new auto-similarity result on the Mandelbrot set.

math.DS↗
arXiv2003

Quadratic polynomials and combinatorics of the principal nest

The definition of principal nest is supplemented with a system of frames that make possible the classification of combinatorial types for every level of the nest. As a consequence, we give necessary and sufficient conditions for the admissibility of a type and prove that given a sequence of non-renormalizable finite admissible types, there is a quadratic polynomial whose nest realizes the sequence.

math.DS↗