# Transversality in the setting of hyperbolic and parabolic maps

## TL;DR

This paper extends techniques for analyzing critical relations in holomorphic maps to cases where critical orbits approach hyperbolic or parabolic cycles, revealing conditions for parameter dependence and bifurcation behavior.

## Contribution

It demonstrates that the unfolding method for critical relations applies to hyperbolic and parabolic limits, establishing criteria for multiplier dependence and orbit persistence.

## Key findings

- Multiplier of hyperbolic attracting orbit varies univalently with parameters
- Bifurcations at parabolic points are generically observed
- Periodic orbits with fixed multiplier can persist under perturbations

## Abstract

In this paper we consider families of holomorphic maps defined on subsets of the complex plane, and show that the technique developed in \cite{LSvS1} to treat unfolding of critical relations can also be used to deal with cases where the critical orbit converges to a hyperbolic attracting or a parabolic periodic orbit. As before this result applies to rather general families of maps, such as polynomial-like mappings, provided some lifting property holds. Our Main Theorem states that either the multiplier of a hyperbolic attracting periodic orbit depends univalently on the parameter and bifurcations at parabolic periodic points are generic, or one has persistency of periodic orbits with a fixed multiplier.