The problem of the decomposability of inverse limits of dynamical systems on continua has been of long-standing interest to many researchers in topological dynamics. A cornerstone result by Barge and Martin (1) shows that if a topological dynamical system on a unit interval has positive entropy, then the inverse limit of this system must contain an indecomposable continuum. Since then, Ingram (2), Ye (3), Mouron (4), and others have carried out a number of studies, e.g., investigating maps exhibiting a local periodic behavior of a special kind on arc-like continua, and especially homeomorphisms of such spaces with positive entropy.

A result by Darji and Kato (5) states that the inverse limit of a topological dynamical system on a graph-like continuum with positive entropy must contain an indecomposable continuum. In this talk, we will show that if we consider the Gehman dendrite, then it is possible to construct a system with arbitrarily large entropy whose inverse limit is hereditarily decomposable.