Rescaling limits of complex rational maps

TL;DR

This paper investigates the limits of sequences of complex rational maps as they approach infinity in parameter space, revealing bounds on rescaling limits and providing a complete classification for quadratic cases using non-Archimedean dynamics.

Contribution

It establishes an upper bound on the number of non-postcritically finite rescaling limits for degree d maps and fully characterizes quadratic rational map limits.

Findings

At most 2d-2 rescaling limits for degree d maps.

Complete classification of quadratic rational map rescaling limits.

Utilizes non-Archimedean dynamics tools.

Abstract

We discuss rescaling limits for sequences of complex rational maps in one variable which approach infinity in parameter space.It is shown that any given sequence of maps of degree $d \ge 2$ has at most $2d-2$ dynamically distinct rescaling limits which are not postcritically finite. For quadratic rational maps, a complete description of the possible rescaling limits is given. These results are obtained employing tools from non-Archimedean dynamics.