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Infinitely many closed paths in the graph of Anosov flows

Mario Shannon

Abstract

Given an Anosov flow on a closed 3-manifold, we are interested in the problem of whether or not making non-trivial Fried surgeries along a finite set of periodic orbits can produce a flow equivalent to itself. We show that for some suspension Anosov flows, there exist infinitely many pairs of periodic orbits satisfying this property.

Infinitely many closed paths in the graph of Anosov flows

Abstract

Given an Anosov flow on a closed 3-manifold, we are interested in the problem of whether or not making non-trivial Fried surgeries along a finite set of periodic orbits can produce a flow equivalent to itself. We show that for some suspension Anosov flows, there exist infinitely many pairs of periodic orbits satisfying this property.

Paper Structure

This paper contains 10 sections, 11 theorems, 18 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Fried Surgery on Anosov flows
  3. Graph of Anosov flows
  4. Infinitely many loops of length 2 based at a suspension
  5. $\infty$-many closed paths
  6. Anosov flows on sol-manifolds
  7. Hyperbolic Dehn surgery
  8. $\mathbb{R}$-covered Anosov flows on hyperbolic 3-manifolds
  9. Birkhoff sections
  10. Construction of the paths

Key Result

Theorem 7

Let $(\phi^A,M_A)$ be the suspension flow generated by a hyperbolic matrix $A\in\mathrm{SL}(2,\mathbb{Z})$ with $\mathop{\mathrm{\mathrm{tr}}}\nolimits(A)\geq 3$. Then,

Figures (3)

  • Figure 1: Fried surgery $(\phi,M)\xrightarrow[]{(\gamma,1)}(\psi,N)$.
  • Figure 2: $\mathbb{R}$-covered Anosov flows
  • Figure 3: Surgery $(\phi^A,M_A)\xrightarrow[]{(\gamma,m),(\alpha,-m)}(\phi^{B},M_A)$, $B\simeq A$ or $B\simeq A^{-1}$.

Theorems & Definitions (25)

  • Remark 1
  • Definition 2
  • Remark 3
  • Definition 4
  • Remark 5
  • Theorem 7: Bonatti-Iakovoglou
  • Remark 8
  • Remark 10
  • Theorem 11: Plante, Plante_Anosov_sol-mfls. See also Barbot-Maquera_cod=1_Anosov_actions
  • Theorem 12: Uniqueness of the $\mathbb{T}^2$-fibration
  • ...and 15 more