1991/06/27 by Yunping Jiang, Jiang, Yunping
Mathematics · Physics and Astronomy · #Analytic and geometric function theory #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals #Quantum chaos and dynamical systems
paper · pdf · doi:10.48550/arxiv.math/9201288
openalex publication_date 1991/06/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We study hyperbolic mappings depending on a parameter ε . Each of them has an invariant Cantor set. As ε tends to zero, the mapping approaches the boundary of hyperbolicity. We analyze the asymptotics of the gap geometry and the scaling function geometry of the invariant Cantor set as ε goes to zero. For example, in the quadratic case, we show that all the gaps close uniformly with speed √ ε. There is a limiting scaling function of the limiting mapping and this scaling function has dense jump discontinuities because the limiting mapping is not expanding. Removing these discontinuities by continuous extension, we show that we obtain the scaling function of the limiting mapping with respect to the Ulam-von Neumann type metric.