  Iommi & Kiwi showed that the Lyapunov spectrum of an expanding map need not
be concave, and posed various problems concerning the possible number of
inflection points. In this paper we answer a conjecture of Iommi & Kiwi by
proving that the Lyapunov spectrum of a two branch piecewise linear map has at
most two points of inflection. We then answer a question of Iommi & Kiwi by
proving that there exist finite branch piecewise linear maps whose Lyapunov
spectra have arbitrarily many points of inflection. This approach is used to
exhibit a countable branch piecewise linear map whose Lyapunov spectrum has
infinitely many points of inflection.
