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We study the growth of Dfn(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...
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