### Abstract

We show that the Poincare return map of the sixth Painleve equation is chaotic along almost every loop, called a non-elementary loop, in the domain of definition. For each such map we construct a natural invariant Borel probability measure and establish some dynamical properties of it such as positivity of the entropy, ergodicity, hyperbolicity, and so on. We also give an algorithm to calculate the entropy in terms of a reduced word of the loop. This is a joint work with my research student Takato Uehara.