Every Weak Perron Number is an End-Periodic Stretch Factor

Abstract

Given any weak Perron number $λ$, we construct an end-periodic homeomorphism $f:Σ\rightarrow Σ$ with Handel-Miller stretch factor equal to $λ$ where $Σ$ is a connected infinite-type surface with finitely many ends all accumulated by genus.

Figures (12)

• Figure 1: The piecewise linear homeomorphism $f_0:V \rightarrow H$.
• Figure 2: The homeomorphism $f_1:V_1 \rightarrow H_1$ with colors indicating the image of each region.
• Figure 3: The 2-complex $\Sigma_2$ corresponding to the integer $d=3$.
• Figure 4: Running Example: The piece map $f_0:V \rightarrow H$ corresponding to $M$ in Example \ref{['example: run ex']} and each $\sigma_k$ and $\tau_k$ equal to the identity map. Colors indicate the image of each region.
• Figure 5: Running Example: The four edge digraphs corresponding to the map $f_0$ shown in \ref{['fig:running ex f0']}. The black directed edges are in the edge digraphs, while the light gray edges are the remaining edges in the digraphs corresponding to $M$ for $D_L$ and $D_R$ and corresponding to $M^T$ for $D_T$ and $D_B$.

Theorems & Definitions (42)

• Theorem 2.1: Lind
• Theorem 3.1
• proof
• Remark 3.2
• Example 4.1
• Definition 4.2
• Definition 4.3
• Definition 5.1
• Definition 5.2
• Lemma 5.3