Abstract:
The study of flows in surfaces and higher-dimensional manifolds has
caught the interest of many scientists because of its numerous applications
such as Hamiltonian flows, which arise from energy-preserving dynamical
systems, billiard flows, flows from meteorological models, most notably the
Lorenz model. These flows are equipped with a natural invariant measure μ,
for instance SRB-measures.
The main goal is to have a better understanding of the properties of
these flows, such as hyperbolicity, ergodicity, mixing and weak mixing and,
in chaotic settings, rates of mixing; that is, we would like to investigate the
asymptotic behaviour of the correlation coefficients

ρt(v, w) = ∫ M v · w ◦ f tdμ − ∫ M vdμ ∫ M wdμ ,

for observables v, w chosen from an appropriate Banach space. Since
mixing rates are one of the strongest statistical properties to have,
other statistical laws, such as the Central Limit Theorem (CLT), usually
follow. Therefore, investigate the rates of mixing is a major tool for
proving other ergodic properties.
In this talk we will focus on the geometrical Lorenz flow and construct a
modification of this model to produce a flow with polynomial decay
of correlations. For that, we will change the nature of the saddle fixed point at
the origin for a neutral fixed point and see how this change will ultimately
lead us to prove polynomial decay of correlations for the modified flow.