Entropy of Bernoulli convolutions and uniform exponential growth for linear groups

Abstract

The exponential growth rate of non polynomially growing subgroups of $G{L}_d$ is conjectured to admit a uniform lower bound. This is known for non-amenable subgroups, while for amenable subgroups it is known to imply the Lehmer conjecture from number theory. In this note, we show that it is equivalent to the Lehmer conjecture. This is done by establishing a lower bound for the entropy of the random walk on the semigroup generated by the maps $x\mapsto \lambda \cdot x\pm 1$, where $\lambda$ is an algebraic number. We give a bound in terms of the Mahler measure of $\lambda$. We also derive a bound on the dimension of Bernoulli convolutions.

Added the requirement to Remark 6 that lambda is a unit. This is relevant only in the case of generalised Bernoulli convolutions, when the coefficients are not +/-1. Final accepted version. To appear in J. Anal. Math. 33 pages

References in corpus (3)

• Absolute continuity of Bernoulli convolutions for algebraic parameters
• On the dimension of Bernoulli convolutions
• Absolute continuity of complex Bernoulli convolutions

Cited by in corpus (3)

• On the dimension of Bernoulli convolutions
• A lower bound for the dimension of Bernoulli convolutions
• The Entropy of Cantor--like measures