B\"ottcher-type potential for the secant map

TL;DR

This paper constructs a Böttcher-type holomorphic map for the secant method's dynamical system near a root, extending the potential's modulus and analyzing the Green's function's properties.

Contribution

It introduces a novel Böttcher-type map for the secant method dynamics, extending potential theory tools to this setting.

Findings

The Böttcher-type map is continuous on the entire basin of attraction.

The modulus of the map is real-analytic away from preimages of the fixed point.

The Green's function is pluriharmonic where finite.

Abstract

We present a construction of a B\"ottcher-type holomorphic map for the potential of the secant method dynamical system near a root-type fixed point. The modulus of the B\"ottcher-type map extends to be continuous on the entire basin of attraction of the fixed point, and is real-analytic away from the iterated preimages of the fixed point. Using this construction, we show the associated Green's function for the fixed point is pluriharmonic wherever it is finite.