Abstract:

It has been a general belief that entropy relates to the complexity of the behavior of a dynamical system. However, it has been shown in 2009 by Harańczyk and Kwietniak that exact maps on an interval can attain lower entropy than pure mixing maps. This phenomenon has been demonstrated to hold also for graph maps by Harańczyk, Kwietniak and Oprocha in 2014. We studied an analogous question whether the same assertion holds for dendrites, and specifically, for the Gehman dendrite. It is known by works of Špitalský that for the Gehman dendrite the
infimum of entropies of exact maps is zero and is not attainable. We will present the construction of a family of pure mixing maps on the Gehman dendrite containing maps of arbitrarily small but positive entropy. This shows that the phenomenon "pure mixing forces higher entropy than exactness'' no longer holds for dendrites.