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arXiv · 2508.05847

B\"ottcher-type potential for the secant map

Nicholas Freeman
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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.

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Nicholas Freeman. 2025-08-07. B\"ottcher-type potential for the secant map. https://arxiv.org/abs/2508.05847

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