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Generating uniform spanning trees from conditioned Bienaym´e–Galton–Watson trees
Get PDFAquest article explora la generació d’arbres generadors uniformes (UST), fonamentals en combinatòria i probabilitat, amb aplicacions en teoria de xarxes i física. Utilitzant processos de Bienaymé–Galton–Watson (BGW) condicionats a un nombre fix de vèrtexs, s’introdueix un mètode per generar arbres generadors uniformes i s’examinen propietats estructurals com l’alçada i l’amplada.This report explores uniform spanning tree (UST) generation, essential in combinatorics and probability with applications in network theory and physics. Using conditioned Bienaymé–Galton–Watson (BGW) processes, it introduces a method to generate USTs. Rigorous proofs show that conditioning on a fixed number of vertices ensures uniform distribution and let us examine structural properties like height and width
Density of hyperbolicity in families of complex rational maps
Get PDFUn dels problemes oberts centrals és la densitat d’hiperbolicitat. En aquest treball ho investiguem en la dinàmica complexa unidimensional, i ens concentrem en el cas polinòmic (cas particular d’una funció racional) com a model on els mecanismes principals poden ser exposats i comprovats en detall. La via procedimental és clara: primer, la construcció de peces de puzle en un entorn del conjunt de Julia; segon, l’ús d’aquestes per definir una funció de caixa complexa; i finalment, l’aplicació de teoremes de rigidesa a aquestes. Aquest procés tradueix la informació combinatòria en rigidesa per als polinomis, demostrant que un polinomi no renormalitzable pot ser aproximat per un polinomi hiperbòlic.In this work, we address the fundamental open problem of whether hyperbolic rational maps, the ones for which every critical point lies in the basin of an attracting cycle, are dense in the space of rational maps of the same degree. By this, we mean if any such map can be uniformly approximated on compact sets by hyperbolic ones. Conjecturally, the answer is ’yes’, and this is known as the Density of Hyperbolicity Conjecture. After reviewing key tools from complex dynamics such as puzzle pieces constructions, quasi-conformal conjugacies, Böttcher coordinates and holomorphic motions, we introduce complex box mappings as a natural extension of polynomial-like maps and discuss their rigidity under combinatorial equivalence. Focusing on non-renormalisable polynomials without neutral periodic points, we reproduce, clarify and check the Kozlovski–van Strien result that such polynomials admit approximating hyperbolic maps by constructing dynamically natural box mappings and applying topological and rigidity results. In conclusion, we outline how this framework, with a careful setting, promises to extend beyond the polynomial case to prove density of hyperbolicity in broader families such as Newton and McMullen maps, thereby sketching a clear path for future advances in complex dynamics
