How Many Inflections are There in the Lyapunov Spectrum?

Abstract

Iommi and Kiwi (J Stat Phys 135:535–546, 2009) 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 in Iommi and Kiwi (2009) 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 in Iommi and Kiwi (2009) 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.

Counting the Lyapunov inflections in piecewise linear systems*

“…Combining theorems 2.2 and 2.4 together, we get the most economical number of branches to observe four Lyapunov inflections for a piecewise linear expanding map. This gives the answer to [JPV,question 6.5] in case of n = 4. Considering theorem 2.4, we can sharpen the general upper bound in [JPV,corollary 6.6], while providing a lower bound by theorem 2.1.…”

“…Remark 2.3. Considering the results [IK, theorem A] and [JPV,theorem 1.3], this theorem weakens one's expectation in counting more number of Lyapunov inflections for piecewise linear maps with increasing branches, which is quite a surprise to us.…”

“…They conjecture that the spectrum has at most two Lyapunov inflections for conformal expanding maps with two branches. The answer is yes for piecewise linear maps, according to [JPV,theorem 1.3], but remains open in the case of nonlinear maps. Another question is on the bound of number of the Lyapunov inflections for the whole family of finitely branched expanding maps.…”

“…Another question is on the bound of number of the Lyapunov inflections for the whole family of finitely branched expanding maps. Jenkinson, Pollicott and Vytnova have given examples to show that, even in the piecewise linear case, the number of Lyapunov inflections is unbounded, see [JPV,theorem 1.4]. To witness a small even number of Lyapunov inflections, the examples of maps in [JPV] cost relatively large numbers of branches.…”

“…To witness a small even number of Lyapunov inflections, the examples of maps in [JPV] cost relatively large numbers of branches. Then they ask about the most economical number of branches in order to observe a given even number of Lyapunov inflections for piecewise linear maps, see [JPV,question 6.5]. We will decide the most economical number of branches to observe four Lyapunov inflections in this work.…”