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Irrational rotations and 2-filling rays

Lvzhou Chen, Alexander J. Rasmussen

TL;DR

The paper studies skew products T over irrational circle rotations to understand orbit density phenomena and uses these dynamics to build geodesic laminations and exotic rays on infinite-type surfaces. By renormalizing via a sequence of first-return maps with carefully chosen continued fraction data for α, it proves that the forward orbit of (1/2,s) is dense in S^1×Z for every s, while individual orbits can still be non-dense, enabling controlled boundary behavior. These dynamical results are then leveraged to construct laminations with dense leaves and to produce infinite cliques of 2-filling rays based at isolated punctures on infinite-type surfaces, with implications for Bavard–Walker loop graphs. The work connects Rauzy–Veech-type induction to laminations and demonstrates the existence of exotic boundary-like structures in the loop graph setting, enriching the understanding of infinite-type surface dynamics and graph boundaries.

Abstract

We study a skew product transformation associated to an irrational rotation of the circle [0,1]/~. This skew product keeps track of the number of times an orbit of the rotation lands in the two complementary intervals of {0,1/2} in the circle. We show that under certain conditions on the continued fraction expansion of the irrational number defining the rotation, the skew product transformation has certain dense orbits. This is in spite of the presence of numerous non-dense orbits. We use this to construct laminations on infinite type surfaces with exotic properties. In particular, we show that for every infinite type surface with an isolated planar end, there is an infinite clique of 2-filling rays based at that end. These 2-filling rays are relevant to Bavard--Walker's loop graphs.

Irrational rotations and 2-filling rays

TL;DR

The paper studies skew products T over irrational circle rotations to understand orbit density phenomena and uses these dynamics to build geodesic laminations and exotic rays on infinite-type surfaces. By renormalizing via a sequence of first-return maps with carefully chosen continued fraction data for α, it proves that the forward orbit of (1/2,s) is dense in S^1×Z for every s, while individual orbits can still be non-dense, enabling controlled boundary behavior. These dynamical results are then leveraged to construct laminations with dense leaves and to produce infinite cliques of 2-filling rays based at isolated punctures on infinite-type surfaces, with implications for Bavard–Walker loop graphs. The work connects Rauzy–Veech-type induction to laminations and demonstrates the existence of exotic boundary-like structures in the loop graph setting, enriching the understanding of infinite-type surface dynamics and graph boundaries.

Abstract

We study a skew product transformation associated to an irrational rotation of the circle [0,1]/~. This skew product keeps track of the number of times an orbit of the rotation lands in the two complementary intervals of {0,1/2} in the circle. We show that under certain conditions on the continued fraction expansion of the irrational number defining the rotation, the skew product transformation has certain dense orbits. This is in spite of the presence of numerous non-dense orbits. We use this to construct laminations on infinite type surfaces with exotic properties. In particular, we show that for every infinite type surface with an isolated planar end, there is an infinite clique of 2-filling rays based at that end. These 2-filling rays are relevant to Bavard--Walker's loop graphs.
Paper Structure (4 sections, 17 theorems, 41 equations, 9 figures)

This paper contains 4 sections, 17 theorems, 41 equations, 9 figures.

Table of Contents

  1. Introduction
  2. Proof of Theorem \ref{['theorem:skewprod']}
  3. Background on laminations and rays
  4. Laminations

Key Result

Theorem 1.1

Suppose that the continued fraction expansion $\alpha=[0;a_1,a_2,\ldots]$ satisfies that $a_1\geq 5$ is odd and $a_n\geq 6$ is even for every $n>1$. Then, for any $s\in \mathbb Z$, the (forward) orbit $\{T^n(1/2,s)\}_{n=0}^\infty$ is dense in $S^1\times \mathbb Z$.

Figures (9)

  • Figure 1: The decomposition of $J$ into $J_i$'s when $b=2n+1=5$ for $n=2$ and the orbit of $J^\mathrm{new}$ under iterations of $t_J$, for $\beta>0$
  • Figure 2: The decomposition of $J$ into $J_i$'s when $b=2n+1=5$ for $n=2$ and the orbit of $J^\mathrm{new}$ under iterations of $t_J$, for $\beta<0$.
  • Figure 3: Left: two singular leaves that split after passing through a splitting singularity; middle: two singular leaves that merge after passing through a merging singularity; right: a singular leaf (indicated by the arrows) passes through three singularities and always splits to the left and merges from the left.
  • Figure 4: A weighted train track $\tau$. The second return map to a horizontal interval in the rectangle $R$ of the union of foliated rectangles $F$ is a rotation by $\alpha$.
  • Figure 5: The infinite cyclic cover $\widetilde{\tau}$ of $\tau$.
  • ...and 4 more figures

Theorems & Definitions (34)

  • Theorem 1.1
  • Corollary 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Proposition 2.1
  • proof : Proof of Theorem \ref{['theorem:skewprod']} assuming Proposition \ref{['prop:firstreturns']}
  • proof : Proof of Corollary \ref{['cor: main']}
  • Lemma 2.5
  • proof
  • Lemma 2.6
  • ...and 24 more