Asymptotic differentiable structure on Cantor set
The subject of the study are unimodal mappings of the interval for which the critical point either escapes to infinity (hyperbolic case) or is mapped to a fixed point in the second iterate (subhyperbolic case). In both cases the map exhibits full chaos. In the hyperbolic case, under sufficient smoothness conditions, the dynamics is expanding. However, the work only assumes \(C^{1+\alpha}\) smoothness and expansion is imposed by the condition that diameters of preimages of the natural Markov partition shrink exponentially fast. In the hyperbolic case, the scaling functions of resulting invariant Cantor sets are studied. Families of hyperbolic maps are then considered tending to a subhyperbolic one. A connection, rather a complicated one, is found between the limits of scaling functions and the orbifold structure of the limiting subhyperbolic map.
scientific article; zbMATH DE number 459112