LI-YORKE CHAOS ALMOST EVERYWHERE: ON THE PERVASIVENESS OF DISJOINT EXTREMALLY SCRAMBLED SETS

We show that there exists a continuous function from the unit Lebesgue interval to itself such that for any epsilon >= 0 and any natural number k, any point in its domain has an epsilon-neighbourhood which, when feasible, contains k mutually disjoint extremally scrambled sets of identical Lebesgue measure, homeomorphic to each other. This result enables a satisfying generalisation of Li-Yorke (topological) chaos and suggests an open (difficult) problem as to whether the result is valid for piecewise linear functions.