On the asymptotic expansion of maps with disconnected Julia set

Abstract

We study the asymptotic expansion of smooth one-dimensional maps. We give an example of an interval map for which the optimal shrinking of components exponential rate is not attained for any neighborhood of a certain fixed point in the boundary of a periodic Fatou component. We prove a general result asserting that, when this happens the components do shrink exponentially, although the rate is not the optimal one. Finally, we give an example of a polynomial with real coefficients, such that all its critical points in the complex plane are real, and such that its asymptotic expansion as a complex map is strictly smaller than its asymptotic expansion as a real map.

Cited by in corpus (4)

• Geometric pressure for multimodal maps of the interval
• Large deviation principles of one-dimensional maps for Hölder continuous potentials
• Thermodynamic formalism of interval maps for upper semi-continuous potentials: Makarov-Smirnov's formalism
• Interval maps quasi-symmetrically conjugated to a piecewise affine map