The Slope Problem in Discrete Iteration

TL;DR

This paper revisits the slope problem in holomorphic dynamics within the unit disk, comparing discrete and continuous cases, and explores the structure of the set of slopes for parabolic functions, providing new insights and examples.

Contribution

It introduces new results on the set of slopes in the discrete parabolic case and connects these findings with the continuous setting, offering novel examples and analysis.

Findings

The set of slopes in the discrete parabolic case is a closed interval independent of initial point.

Any such interval can be realized in the continuous setting.

Examples of functions with non-trivial slope sets are constructed.

Abstract

The slope problem in holomorphic dynamics in the unit disk goes back to Wolff in 1929. However, there have been several contributions to this problem in the last decade. In this article the problem is revisited, comparing the discrete and continuous cases. Some advances are derived in the discrete parabolic case of zero hyperbolic step, showing that the set of slopes has to be a closed interval which is independent of the initial point. The continuous setting is used to show that any such interval is a possible example. In addition, the set of slopes of a family of parabolic function is discussed, leading to examples of functions with some regularity whose set of slopes is non-trivial.