# Dynamics on immediate basins for parabolic Newton maps

## TL;DR

This paper investigates the structure of immediate basins in parabolic Newton maps, revealing unique invariant accesses and the dynamics of attraction and repulsion near parabolic fixed points.

## Contribution

It establishes the existence of a unique dynamically attracting access in parabolic immediate basins and characterizes the behavior of other accesses.

## Key findings

- Every parabolic immediate basin contains invariant accesses to infinity.
- There exists a unique access where dynamics are attracted to the fixed point.
- Other accesses, if any, exhibit repelling dynamics.

## Abstract

Dynamics on parabolic immediate basins for rational Newton maps of entire functions have been studied. It is proved that every parabolic immediate basin contains invariant accesses to the parabolic fixed point at infinity. Moreover, among these accesses there exists a unique dynamically defined access where dynamics are attracted towards the parabolic fixed point, whereas for other accesses, if there is any, the dynamics are repelled.