Geometric limits of mixed families

Abstract

Letting f and h denoting rational maps of degree at least 2 and 1 respectively, and denote by $f^n$ the nth iterate of f. We give upper and lower bounds on the set of accumulation points of the Julia sets $\mathcal{J}(h\circ f^n)$ with respect to the Hausdorff metric on non-empty compact subsets of the sphere.

Keywords:

• Julia set
• geometric limit
• Hausdorff convergence
• rational iteration

2020 Mathematics Subject Classification:

• 32H50

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

The authors would like to thank the Danish Council for Independent Research – Natural Sciences for support via the grant DFF–1026–00267B.