  We consider stochastic processes arising from dynamical systems by evaluating
an observable function along the orbits of the system. The novelty is that we
will consider observables achieving a global maximum value (possible infinite)
at multiple points with special emphasis for the case where these maximal
points are correlated or bound by belonging to the same orbit of a certain
chosen point. These multiple correlated maxima can be seen as a new mechanism
creating clustering. We recall that clustering was intimately connected with
periodicity when the maximum was achieved at a single point. We will study this
mechanism for creating clustering and will address the existence of limiting
Extreme Value Laws, the repercussions on the value of the Extremal Index, the
impact on the limit of Rare Events Points Processes, the influence on
clustering patterns and the competition of domains of attraction. We also
consider briefly and for comparison purposes multiple uncorrelated maxima. The
systems considered include expanding maps of the interval such as Rychlik maps
but also maps with an indifferent fixed point such as Manneville-Pommeau maps.
