  In the context of Wilsonian Renormalization, renormalization group (RG) flows
are a set of differential equations that defines how the coupling constants of
a theory depend on an energy scale. These equations closely resemble
thermodynamical equations and how thermodynamical systems are related to
temperature. In this sense, it is natural to look for structures in the flows
that show a thermodynamics-like behaviour. The mathematical theory to study
these equations is called Dynamical Systems, and applications of that have been
used to study RG flows. For example, the classical Zamolodchikov's C-Theorem
and its higher-dimensional counterparts, that show that there is a
monotonically decreasing function along the flow and it is a property that
resembles the second-law of thermodynamics, is related to the Lyapunov function
in the context of Dynamical Systems. It can be used to rule out exotic
asymptotic behaviours like periodic flows (also known as limit cycles). We also
study bifurcation theory and index theories, which have been proposed to be
useful in the study of RG flows, the former can be used to explain couplings
crossing through marginality and the latter to extract global information about
the space the flows lives in. In this dissertation, we also look for
applications in holographic RG flows and we try to see if the structural
behaviours in holographic theories are the same as the ones in the dual field
theory side.
