Mahler measure of polynomial iterates

Abstract

Granville recently asked how the Mahler measure behaves in the context of polynomial dynamics. For a polynomial 𝑓 (𝑧) = 𝑧 𝑑 + . . . ∈ C[𝑧 ], deg( 𝑓 ) ≥ 2, we show that the Mahler measure of the iterates 𝑓 𝑛 grows geometrically fast with the degree 𝑑 𝑛 , and find the exact base of that exponential growth. This base is expressed via an integral of log + | 𝑧 | with respect to the invariant measure of the Julia set for the polynomial 𝑓 . Moreover, we give sharp estimates for such an integral when the Julia set is connected.