Browsing by Author "Pérez, Rodrigo A."

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    Accessibility of the Boundary of the Thurston Set
    (Taylor & Francis, 2023) Silvestri, Stefano; Pérez, Rodrigo A.; Mathematical Sciences, School of Science
    Consider two objects associated to the Iterated Function System (IFS) {1+𝜆⁢𝑧,−1+𝜆⁢𝑧}: the locus ℳ of parameters 𝜆∈𝔻∖{0} for which the corresponding attractor is connected; and the locus ℳ0 of parameters for which the related attractor contains 0. The set ℳ can also be characterized as the locus of parameters for which the attractor of the IFS {1+𝜆⁢𝑧,𝜆⁢𝑧,−1+𝜆⁢𝑧} contains 𝜆−1. Exploiting the asymptotic similarity of ℳ and ℳ0 with the respective associated attractors, we give sufficient conditions on 𝜆∈∂ℳ or ∂ℳ0 to guarantee it is path accessible from the complement 𝔻∖ℳ.
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    Scrambled Vandermonde convolutions of Gaussian polynomials
    (Elsevier, 2022-12) Aspenburg, Magnus; Pérez, Rodrigo A.; Mathematical Sciences, School of Science
    It is well known that Gaussian polynomials (i.e., q-binomials) describe the distribution of the area statistic on monotone paths in a rectangular grid. We introduce two new statistics, corners and c-index; attach "ornaments" to the grid; and re-evaluate these statistics, in order to argue that all scrambled versions of the c-index statistic are equidistributed with area. Our main result is a representation of the generating function for the bi-statistic (c-index; corners) as a two-variable Vandermonde convolution of the original Gaussian polynomial. The proof relies on explicit bijections between differently ornated paths.