  Local correlation entropy, introduced by Takens in 1983, represents the
exponential decay rate of the relative frequency of recurrences in the
trajectory of a point, as the embedding dimension grows to infinity. In this
paper we study relationship between the supremum of local correlation entropies
and the topological entropy. For dynamical systems on graphs we prove that the
two quantities coincide. Moreover, there is an uncountable set of points with
local correlation entropy arbitrarily close to the topological entropy. On the
other hand, we construct a strictly ergodic subshift with positive topological
entropy having all local correlation entropies equal to zero. As a necessary
tool, we derive an expected relationship between the local correlation
entropies of a system and those of its iterates.
