Iterated Monodromy Group With Non-Martingale Fixed-Point Process

Abstract

We construct families of rational functions $f: \mathbb{P}^1_k \rightarrow \mathbb{P}^1_k$ of degree $d \geq 2$ over a perfect field $k$ with non-martingale fixed-point processes. Then for any normal variety $X$ over $\mathbb{P}_{\bar{k}}^N$, we give conditions on $f: X \rightarrow X$ to guarantee that the associated fixed-point process is a martingale. This work extends the previous work of Bridy, Jones, Kelsey, and Lodge on martingale conditions and answers their question on the existence of a non-martingale fixed-point process associated with the iterated monodromy group of a rational function.

Figures (4)

• Figure 1: Ramification Portraits of $g \circ \psi$.
• Figure 2: We get a level $4$ tree by decomposing $\phi$ as $g \circ h$.
• Figure 3: Tree of cosets
• Figure :

Theorems & Definitions (48)

• Definition 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5: Theorem \ref{['high']}
• Definition 2.1
• Lemma 2.2: Burnside's Lemma
• Lemma 2.3: Coset Burnside's Lemma
• proof
• Lemma 2.4: Abhyankar's Lemma, codes