This dissertation is in the area of complex dynamics, more specifically focused on the iteration of rational functions. Given a well-chosen family of rational functions, parameterized by a complex parameter, we are especially interested in regularity properties of the Hausdorff dimension of Julia sets of these polynomials considered as a function of the parameters. In this dissertation I deal with a family of polynomials of degree at least 3 depending in a holomorphic way on a parameter, focusing on the point where the dynamics and topology of the polynomials drastically change. In such a context proving continuity is quite challenging while real analyticity will most likely break. Our approach will, on the one hand, build on the existing methods of proving continuity of Hausdorff dimension, primarily based on proving continuity, in the weak* topology of measures on the Riemann sphere, of canonical conformal measures, but will also require methods which, up to my best knowledge, have not been implemented anywhere yet. Our main result gives a surprising example where the Hausdorff dimension of the Julia set is continuous in the parameter, but where the Julia set itself is not.

The objective of this work is the study of phase transitions on the context of Groupoids and their C*-Algebras. The main result of this dissertation is due to Klaus Thomsen in [Tho17], which investigates the connection between conformal measures in the classical formalism and KMS-states in the quantum formalism. The phase transition in the quantum setting is a consequence of this connection between both formalisms and the fact that on the classical setting it was known examples of continuous potentials that show the phenomena of phase transition. The potential used was introduced by Hofbauer [Hof77], an example that shows, dierently from potential of summable variations, potentials only continuous can exhibit phase transition. / O objetivo deste trabalho é o estudo do fenômeno de transição de fase no contexto de Grupóides e suas C*-álgebras. O resultado principal é devido a Klaus Thomsen em [Tho17], que explora a conexão entre medidas conformes no formalismo clássico e estados KMS do contexto quântico. A transição de fase no caso quântico é consequência desta ligação entre os dois formalismos e do fato de que no setting clássico eram conhecidos exemplos de potenciais contínuos que apresentam o fenômeno de transição de fase. O potencial utilizado é aquele introduzido por Hofbauer [Hof77], um exemplo que mostra que, diferentemente de potenciais de variação somável, potenciais apenas contínuos podem apresentar transição de fase.