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DCDS

We improve Mather's proof on the existence of the
connecting orbit around rotation number zero
(Proposition 8.1 in [7]) in this paper.
Our new proof not only assures the existences of the
connecting orbit, but also gives a quantitative
estimation on the diffusion time.

DCDS

We analyze parametrized families of multimodal $1D$ maps that arise as singular limits
of parametrized families of rank one maps. For a generic $1$-parameter family of such
maps that contains a Misiurewicz-like map, it has been shown that in a neighborhood of
the Misiurewicz-like parameter, a subset of parameters of positive Lebesgue measure
exhibits nonuniformly expanding dynamics characterized by the existence of a positive
Lyapunov exponent and an absolutely continuous invariant measure. Under a mild
combinatoric assumption, we prove that each such parameter is an accumulation point of
the set of parameters admitting superstable periodic sinks.

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