Dr Polina Vytnova

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.

In this note we present an algorithm to obtain a uniform lower bound on Hausdorff dimension of the stationary measure of an affine iterated function scheme with similarities, the best known example of which is Bernoulli convolution. The Bernoulli convolution measure $\mu_\lambda$ is the probability measure corresponding to the law of the random variable $\xi = \sum_{k=0}^\infty \xi_k\lambda^k$, where $\xi_k$ are i.i.d. random variables assuming values $-1$ and $1$ with equal probability and $\frac12 < \lambda < 1$. In particular, for Bernoulli convolutions we give a uniform lower bound $\dim_H(\mu_\lambda) \geq 0.96399$ for all $\frac12

In this short note we describe a simple but remarkably effective method for rigorously estimating Lyapunov exponents for expanding maps of the interval. We illustrate the applicability of this method with some standard examples.