Counting the Lyapunov inflections in piecewise linear systems.

Following the pioneering work of Iommi-Kiwi and Jenkinson-Pollicott-Vytnova, we continue to study the inflection points of the Lyapunov spectrum in this work. We prove that for any 3-branch piecewise linear expanding map on an interval, the number of its Lyapunov inflections is bounded above by 2. Then we continue to show that, there is a 4-branch piecewise linear expanding map, such that its Lyapunov spectrum has exactly 4 inflection points. These results give an answer to a question by Jenkinson-Pollicott-Vytnova on the least number of branches needed to observe 4 inflections in the Lyapunov spectrum of piecewise linear maps. In the general case, we give upper bound on the number of Lyapunov inflections for any n-branch piecewise linear expanding maps, and construct a family of n-branch piecewise linear expanding maps with 2n-4 Lyapunov inflections. We also consider the number of Lyapunov inflections of piecewise linear maps in words of the essential branch number in this work. There are some results on distributions of the Lyapunov inflections of piecewise linear maps through out the work, in case of their existence.