DCDS
The diffusion time of the connecting orbit around rotation number zero for the monotone twist maps
Qiudong Wang
Discrete & Continuous Dynamical Systems - A 2000, 6(2): 255-274 doi: 10.3934/dcds.2000.6.255
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.
keywords: diffusion time. quantitative stimation Connecting orbit
DCDS
Periodic attractors versus nonuniform expansion in singular limits of families of rank one maps
William Ott Qiudong Wang
Discrete & Continuous Dynamical Systems - A 2010, 26(3): 1035-1054 doi: 10.3934/dcds.2010.26.1035
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.
keywords: nonuniformly expanding map admissible family of $1D$ maps parametrized family of maps rank one map absolutely continuous invariant measure. periodic attractor singular limit

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