Decay of geometry for Fibonacci critical covering maps of the circle

Abstract We study the growth of D f n ( f ( c ) ) when f is a Fibonacci critical covering map of the circle with negative Schwarzian derivative, degree d ⩾ 2 and critical point c of order l > 1 . As an application we prove that f exhibits exponential decay of geometry if and only if l ⩽ 2 , and in this case it has an absolutely continuous invariant probability measure, although not satisfying the so-called Collet–Eckmann condition.