Almost equivalence of suspension Anosov flows

Abstract

We provide a written proof of a result due to H. Minakawa, which states that all suspension Anosov flows generated by hyperbolic matrices with positive trace are pairwise almost equivalent. The proof relies on constructing, for any given suspension flow, a genus-one Birkhoff section whose first-return map has fewer fixed points than the original map. We improve Minakawa's result by explicitly calculating the first return map onto this section, which leads to explicit bounds on the distances between suspension Anosov flows within the graph of Anosov flows.

Figures (6)

• Figure 1: Fried desingularization.
• Figure 2: The polygons $\widetilde{P}_{XW}$ (left) and $\widetilde{P}_{YW}$ in $\mathbb{R}^2$.
• Figure 3: The parallelogram $P$ (green) inside the torus $\mathbb{T}^2$. The torus is represented as the square $[0,1]^2$ (left) and the square $[-1/2, 1/2]^2$ (right). Red dots are fixed points of $XW$, with $XW = \left(38411\right)$ in this example. The action of $XW$ maps $r_1$ to $r_0$ and $s_1$ to $s_0$. If $W$ is not of the form $X^mY^n$ or $Y^nX^m$, the point $N$ lies in the interior of the dotted triangle; otherwise, it lies on its boundary.
• Figure 4: The construction of the immersed pair of pants $\mathbb{P}^\perp$ (center) and its smoothing $\mathbb{P}$ (on the right).
• Figure 5: Fried desingularization of $\partial\mathbb{T}^2_{2/3}$ (green) and $\partial\mathbb{P}$ (black) along on the components $T_M$ (left), $T_O$ (center) and $T_N$ (right) of $\partial\mathbb{T}^3_{XW, \Gamma}$.

Theorems & Definitions (22)

• Theorem A: Minakawa Minakawa
• Theorem B: Minakawa Minakawa
• Theorem C
• Corollary 1.2
• Definition 1.3
• Theorem D
• Definition 2.1
• Lemma 2.2
• proof
• Lemma 2.3