On the Combinatorics of Fibonacci-Like Non-renormalizable Maps

Abstract

In this paper, we extend Fibonacci unimodal map to a wider class. We describe the combinatorial property of these maps by first return map and principal nest. We give the sufficient and necessary condition for the existence of this class of maps. Moreover, for maps with ’bounded combinatorics’, we prove that they have no absolutely continuous invariant probability measure when the critical order $\ell $ is sufficiently large; for maps with reluctantly recurrent critical point, we prove they have absolutely continuous invariant probability measure whenever the critical order $\ell >1$.

Keywords

Fibonacci / Principal nest / Invariant measure / Generalized renormalization