Abstract
We discuss one parameter families of unimodal maps, with negative Schwarzian derivative, unfolding a saddle-node bifurcation. We show that there is a parameter set of positive but not full Lebesgue density at the bifurcation, for which the maps exhibit absolutely continuous invariant measures which are supported on the largest possible interval. We prove that these measures converge weakly to an atomic measure supported on the orbit of the saddle-node point. Using these measures we analyze the intermittent time series that result from the destruction of the periodic attractor in the saddle-node bifurcation and prove asymptotic formulae for the frequency with which orbits visit the region previously occupied by the periodic attractor.
Highlights
By Jakobson’s celebrated work [Jak81], the logistic family x → μx(1 − x), x ∈ [0, 1], admits absolutely continuous invariant measures (a.c.i.m.’s) for μ from a set of positive measure
A different argument for this result was given by Benedicks and Carleson [BenCar85], [BenCar91]. Their reasoning was generalized to unfoldings {fγ} of unimodal Misiurewicz maps f0, with eventually periodic critical point c
It was shown that the bifurcation value γ = 0 is a density point of the set of parameter values for which fγ admits an absolutely continuous invariant measure [MelStr93], [ThiTreYou94]
Summary
By Jakobson’s celebrated work [Jak81], the logistic family x → μx(1 − x), x ∈ [0, 1], admits absolutely continuous invariant measures (a.c.i.m.’s) for μ from a set of positive measure. A different argument for this result was given by Benedicks and Carleson [BenCar85], [BenCar91] Their reasoning was generalized to unfoldings {fγ} of unimodal Misiurewicz maps f0, with eventually periodic critical point c (and possessing negative Schwarzian derivative). We establish that a saddle-node bifurcation value occurs as a point of positive density of the parameter set for which there are absolutely continuous invariant measures. Both periodic attractors and a.c.i.m’s occur with positive density at the bifurcation value This part of our paper is related to results of M.J. Costa [Cos03] on absolutely continuous measures in certain families of unimodal maps near a saddle-node bifurcation of a fixed point. We are grateful for the detailed comments from the referees that helped improve the presentation of the paper
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