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Gaps of saddle connection directions for some branched covers of tori Anthony Sanchez asanch33@uw.edu West Coast Dynamics Seminar May 14 th , 2020 Translation surfaces A translation surface is a collection of polygons with edge identifications

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Gaps of saddle connection directions for some branched covers of tori

Anthony Sanchez asanch33@uw.edu West Coast Dynamics Seminar

May 14th, 2020

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Translation surfaces

A translation surface is a collection of polygons with edge identifications given by translations.

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Translation surfaces

A translation surface is a collection of polygons with edge identifications given by translations. Torus

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Translation surfaces

A translation surface is a collection of polygons with edge identifications given by translations. Torus

  • Genus 1
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Translation surfaces

A translation surface is a collection of polygons with edge identifications given by translations. Torus

  • Genus 1
  • Flat geometry everywhere.
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Octagon

Regular Octagon:

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Octagon

Regular Octagon:

  • Genus 2
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Octagon

Regular Octagon:

  • Genus 2
  • Single cone point of angle 6𝜌
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Doubled slit torus construction

Take a flat torus and mark two points

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Take an identical copy of the twice-marked torus

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Cut a slit between the marked points

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Glue opposite sides of the slit together

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Doubled Slit Torus

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Doubled Slit Torus

Genus 2 surface

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Doubled Slit Torus

Genus 2 surface 2 cone type singularities of angle 4Ο€

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Doubled Slit Torus

Genus 2 surface 2 cone type singularities of angle 4Ο€

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Doubled Slit Torus

Genus 2 surface 2 cone type singularities of angle 4Ο€

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Doubled Slit Torus

Genus 2 surface 2 cone type singularities of angle 4Ο€

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Doubled Slit Torus

Genus 2 surface 2 cone type singularities of angle 4Ο€

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Why doubled slit tori?

(Topology) Are a natural construction of a higher genus surface from genus 1 surfaces.

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Why doubled slit tori?

(Topology) Are a natural construction of a higher genus surface from genus 1 surfaces. (Dynamics) First higher genus surface with minimal but not uniquely ergodic straight-line flow.

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Why doubled slit tori?

(Topology) Are a natural construction of a higher genus surface from genus 1 surfaces. (Dynamics) First higher genus surface with minimal but not uniquely ergodic straight-line flow. (Geometry) Are examples of translation surfaces.

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Translation structure

Embedding into complex plane endows the surface with a Riemann surface structure π‘Œ

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Translation structure

Embedding into complex plane endows the surface with a Riemann surface structure π‘Œ and the holomorphic differential 𝑒𝑨.

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Translation surfaces

More generally any pair π‘Œ, πœ• where π‘Œ is a Riemann surface and πœ• is a non-zero holomorphic differential is called a translation surface.

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Translation surfaces

More generally any pair π‘Œ, πœ• where π‘Œ is a Riemann surface and πœ• is a non-zero holomorphic differential is called a translation surface. The holomorphic differential allows us to measure lengths and gives a sense of direction.

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We are interested in paths on doubled slit tori

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A saddle connection is a straight-line trajectory starting and ending at a cone type singularity.

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Associated to each saddle connection is the holonomy vector.

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Associated to each saddle connection is the holonomy vector. Χ¬

𝛿 𝑒𝑨 = 4 + 𝑗

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Associated to each saddle connection is the holonomy vector. Χ¬

𝛿 𝑒𝑨 = 4 + 𝑗 or 4

1

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πœ€

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Χ¬

πœ€ 𝑒𝑨 = 1 + 0𝑗 or 1

πœ€

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Χ¬

𝛿 𝑒𝑨 = 4 + 𝑗 or 4

1 and Χ¬

πœ€ 𝑒𝑨 = 1 + 0𝑗 or 1

πœ€

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Let π›­πœ• denote the set of all holonomy vectors.

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Let π›­πœ• denote the set of all holonomy vectors.

π›­πœ•

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Let π›­πœ• denote the set of all holonomy vectors. Veech: π›­πœ• is a discrete subset!

π›­πœ•

Discreteness

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How random are the holonomy vectors?

π›­πœ• π‘Œ, πœ•

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How random are the holonomy vectors?

π›­πœ• π‘Œ, πœ•

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Angles as a test of randomness

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Angles as a test of randomness

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Angles as a test of randomness

  • Masur: angles are dense
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Angles as a test of randomness

  • Masur: angles are dense
  • Vorobets: angles are equidistributed

for almost every translation surface

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Angles as a test of randomness

  • Masur: angles are dense
  • Vorobets: angles are equidistributed

for almost every translation surface

  • Eskin-Marklof-Morris: angles are

equidistributed for covers of lattices surfaces

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Upshot: Saddle connections appear to behave randomly at first glance.

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A second test of randomness

A second test of randomness is to consider gaps of sequences.

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A second test of randomness

A second test of randomness is to consider gaps of sequences. We consider slopes of saddle connections instead of angles.

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Slopes of holonomy vectors

Let π‘‡π‘šπ‘π‘žπ‘“π‘‘π‘† π›­πœ• denote the slopes in an eighth sector up to length 𝑆.

π›­πœ•

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Slopes of holonomy vectors

Let π‘‡π‘šπ‘π‘žπ‘“π‘‘π‘† π›­πœ• denote the slopes in an eighth sector up to length 𝑆. π‘‡π‘šπ‘π‘žπ‘“π‘‘π‘† π›­πœ• = 𝑑0 = 0 < 𝑑1 < β‹― < 𝑑𝑂(𝑆) where 𝑂 𝑆 = |π‘‡π‘šπ‘π‘žπ‘“π‘‘π‘† π›­πœ• |.

π›­πœ•

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Slopes of holonomy vectors

Let π‘‡π‘šπ‘π‘žπ‘“π‘‘π‘† π›­πœ• denote the slopes in an eighth sector up to length 𝑆. π‘‡π‘šπ‘π‘žπ‘“π‘‘π‘† π›­πœ• = 𝑑0 = 0 < 𝑑1 < β‹― < 𝑑𝑂(𝑆) where 𝑂 𝑆 = |π‘‡π‘šπ‘π‘žπ‘“π‘‘π‘† π›­πœ• |. Eskin-Masur showed 𝑂 𝑆 ~ 𝑆2.

π›­πœ•

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Gaps of holonomy vectors

Consider the gaps of slopes π»π‘π‘žπ‘‘π‘† π›­πœ• = (𝑑𝑗 βˆ’ π‘‘π‘—βˆ’1)| 𝑗 = 1, … , 𝑂(𝑆)

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Gaps of holonomy vectors

Consider the gaps of slopes π»π‘π‘žπ‘‘π‘† π›­πœ• = 𝑆2(𝑑𝑗 βˆ’ π‘‘π‘—βˆ’1)| 𝑗 = 1, … , 𝑂(𝑆)

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Gaps of holonomy vectors

Consider the gaps of slopes π»π‘π‘žπ‘‘π‘† π›­πœ• = 𝑆2(𝑑𝑗 βˆ’ π‘‘π‘—βˆ’1)| 𝑗 = 1, … , 𝑂(𝑆) What can we say about the distribution of gaps?

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Gap distribution

The gap distribution is given by π»π‘π‘žπ‘‘π‘† π›­πœ•

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Gap distribution

The gap distribution is given by π»π‘π‘žπ‘‘π‘† π›­πœ• ∩ 𝐽

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Gap distribution

The gap distribution is given by |π»π‘π‘žπ‘‘π‘† π›­πœ• ∩ 𝐽|

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Gap distribution

The gap distribution is given by π»π‘π‘žπ‘‘π‘† π›­πœ• ∩ 𝐽 𝑂(𝑆)

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Gap distribution

The gap distribution is given by π‘šπ‘—π‘›

π‘†β†’βˆž

π»π‘π‘žπ‘‘π‘† π›­πœ• ∩ 𝐽 𝑂(𝑆)

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Gap distribution

The gap distribution is given by π‘šπ‘—π‘›

π‘†β†’βˆž

π»π‘π‘žπ‘‘π‘† π›­πœ• ∩ 𝐽 𝑂(𝑆) This measures the proportion of gaps in an interval 𝐽.

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Gap distribution

The gap distribution is given by π‘šπ‘—π‘›

π‘†β†’βˆž

π»π‘π‘žπ‘‘π‘† π›­πœ• ∩ 𝐽 𝑂(𝑆) This measures the proportion of gaps in an interval 𝐽. What can we say about this limit? What do we expect?

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Context from probability

Suppose that π‘Œπ‘— 𝑗=1

∞

are a sequence of IID random variables uniformly distributed on [0,1].

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Context from probability

Suppose that π‘Œπ‘— 𝑗=1

∞

are a sequence of IID random variables uniformly distributed on [0,1].

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Context from probability

Suppose that π‘Œπ‘— 𝑗=1

∞

are a sequence of IID random variables uniformly distributed on [0,1].

π»π‘π‘žπ‘‘{ π‘Œπ‘— 𝑗=1

π‘œ

}

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Context from probability

Suppose that π‘Œπ‘— 𝑗=1

∞

are a sequence of IID random variables uniformly distributed on [0,1].

π»π‘π‘žπ‘‘{ π‘Œπ‘— 𝑗=1

π‘œ

} ∩ 𝐽 π‘œ

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Context from probability

Suppose that π‘Œπ‘— 𝑗=1

∞

are a sequence of IID random variables uniformly distributed on [0,1]. The associated gaps are exponential.

π»π‘π‘žπ‘‘{ π‘Œπ‘— 𝑗=1

π‘œ

}∩𝐽 π‘œ

β†’ Χ¬

I π‘“βˆ’π‘¦ 𝑒𝑦

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The gap distribution of almost every doubled slit torus is not exponential.

Theorem (S. 2020)

π›­πœ• π‘Œ, πœ•

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There exists a density function 𝑔 so that π‘šπ‘—π‘›

π‘†β†’βˆž π»π‘π‘žπ‘‘π‘† π›­πœ• ∩𝐽 𝑂(𝑆)

= Χ¬

𝐽 𝑔 𝑦 𝑒𝑦

for almost every doubled slit torus.

Theorem (S. 2020)

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The gap distribution has a quadratic tail: ΰΆ±

𝑒 ∞

𝑔 𝑦 𝑒𝑦 ~π‘’βˆ’2.

Large gaps

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The gap distribution has a quadratic tail: ΰΆ±

𝑒 ∞

𝑔 𝑦 𝑒𝑦 ~π‘’βˆ’2. Compare with the IID case: ΰΆ±

𝑒 ∞

π‘“βˆ’π‘¦π‘’π‘¦ = π‘“βˆ’π‘’.

Large gaps

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The gap distribution has a quadratic tail: ΰΆ±

𝑒 ∞

𝑔 𝑦 𝑒𝑦 ~π‘’βˆ’2. Compare with the IID case: ΰΆ±

𝑒 ∞

π‘“βˆ’π‘¦π‘’π‘¦ = π‘“βˆ’π‘’.

Large gaps

Thus, large gaps are unlikely, but still much more likely than the random case!

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The gap distribution has support at zero: ΰΆ±

𝜁

𝑔 𝑦 𝑒𝑦 > 0 for every 𝜁 > 0.

Small gaps

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The gap distribution has support at zero: ΰΆ±

𝜁

𝑔 𝑦 𝑒𝑦 > 0 for every 𝜁 > 0.

Small gaps

This is expected since doubled slit tori are not lattice surfaces.

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Higher genus

These surfaces are called symmetric torus covers.

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Symmetric torus covers have the same gap distribution as doubled slit tori.

Higher genus

These surfaces are called symmetric torus covers.

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Other results on gaps of translation surfaces

  • Non-lattice surfaces
  • Lattice surfaces (highly symmetric

translation surfaces)

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Gaps of lattice surfaces

  • Athreya-Cheung (2014) - Torus
  • Athreya-Chaika-Lelievre (2015) -

Golden L

  • Uyanik-Work (2016) - Regular
  • ctagon
  • Taha (2020)- Gluing two regular

(2n+1)-gons

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Gaps of lattice surfaces

  • Athreya-Cheung (2014) - Torus
  • Athreya-Chaika-Lelievre (2015) -

Golden L

  • Uyanik-Work (2016) - Regular
  • ctagon
  • Taha (2020)- Gluing two regular

(2n+1)-gons Characteristics of the gap distributions:

  • No small gaps
  • 2-dimensional parameter space
  • Explicit gap distributions
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Gaps of non-lattice surfaces

Athreya-Chaika (2012) – Generic translation surfaces

  • Gap distribution exists for a.e. translation surface and is the same
  • Non-explicit
  • Small gaps characterize non-lattice surfaces
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Gaps of non-lattice surfaces

Work (2019) – β„‹ 2 Genus 2, single cone point

  • Parameter space 6-dimensional
  • Non-explicit

Athreya-Chaika (2012) – Generic translation surfaces

  • Gap distribution exists for a.e. translation surface and is the same
  • Non-explicit
  • Small gaps characterize non-lattice surfaces
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Gaps of non-lattice surfaces

Work (2019) – β„‹ 2 Genus 2, single cone point

  • Parameter space 6-dimensional
  • Non-explicit

Athreya-Chaika (2012) – Generic translation surfaces

  • Gap distribution exists for a.e. translation surface and is the same
  • Non-explicit
  • Small gaps characterize non-lattice surfaces
  • S. (2020) – Doubled slit tori
  • Parameter space 4-dimensional
  • First explicit gap distribution for

non-lattice surface

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This concludes Part 1

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Part 2: Elements of proof

Anthony Sanchez asanch33@uw.edu

May 14th, 2020

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Elements of the proof

  • Turn gap question into a dynamical

question

  • On return times and affine lattices
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Guiding philosophy

Questions about a fixed translation surface can be understood by considering the dynamics on the space of all translation surfaces.

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Guiding philosophy

Questions about a fixed translation surface can be understood by considering the dynamics on the space of all translation surfaces. Dynamical question on the space of doubled slit tori Gap distribution

  • f a doubled slit

torus

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Translation surfaces β„°

Let β„° denote the set of all doubled slit tori

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The 𝑇𝑀 2, ℝ -action

There is a β€œlinear” action of 𝑇𝑀 2, ℝ on β„°

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The 𝑇𝑀 2, ℝ -action

There is a β€œlinear” action of 𝑇𝑀 2, ℝ on β„°: act on the polygon presentation

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The 𝑇𝑀 2, ℝ -action

There is a β€œlinear” action of 𝑇𝑀 2, ℝ on β„°: act on the polygon presentation

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The 𝑇𝑀 2, ℝ -action

There is a β€œlinear” action of 𝑇𝑀 2, ℝ on β„°: act on the polygon presentation

1 1 1

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The 𝑇𝑀 2, ℝ -action

There is a β€œlinear” action of 𝑇𝑀 2, ℝ on β„°: act on the polygon presentation

1 1 1

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Horocycle flow

Consider the 1-parameter family

β„Žπ‘£ = 1 βˆ’π‘£ 1 : 𝑣 ∈ ℝ

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Horocycle flow

Consider the 1-parameter family

β„Žπ‘£ = 1 βˆ’π‘£ 1 : 𝑣 ∈ ℝ

  • Vertical shear on the plane.
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Horocycle flow

Consider the 1-parameter family

β„Žπ‘£ = 1 βˆ’π‘£ 1 : 𝑣 ∈ ℝ

  • Vertical shear on the plane.
  • This subgroup is of interest because of how it changes slopes.
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Slopes

β„Žπ‘£ 𝑦 𝑧 = 𝑦 𝑧 βˆ’ 𝑣𝑦

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Slopes

β„Žπ‘£ 𝑦 𝑧 = 𝑦 𝑧 βˆ’ 𝑣𝑦

π‘‘π‘šπ‘π‘žπ‘“ β„Žπ‘£ 𝑦 𝑧

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Slopes

β„Žπ‘£ 𝑦 𝑧 = 𝑦 𝑧 βˆ’ 𝑣𝑦

π‘‘π‘šπ‘π‘žπ‘“ β„Žπ‘£ 𝑦 𝑧 = π‘‘π‘šπ‘π‘žπ‘“ 𝑦 𝑧 βˆ’ 𝑣

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Slopes

β„Žπ‘£ 𝑦 𝑧 = 𝑦 𝑧 βˆ’ 𝑣𝑦

π‘‘π‘šπ‘π‘žπ‘“ β„Žπ‘£ 𝑦 𝑧 = π‘‘π‘šπ‘π‘žπ‘“ 𝑦 𝑧 βˆ’ 𝑣

In particular, slope differences are preserved!

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Transversal for doubled slit tori

Consider the transversal for doubled slit tori

𝒳 = πœ•πœ—β„°|π›­πœ• ∩ 0,1 β‰  βˆ…

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Transversal for doubled slit tori

Consider the transversal for doubled slit tori

𝒳 = πœ•πœ—β„°|π›­πœ• ∩ 0,1 β‰  βˆ…

That is, the doubled slit tori that have a short horizontal saddle connection.

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Transversal for doubled slit tori

Consider the transversal for doubled slit tori

𝒳 = πœ•πœ—β„°|π›­πœ• ∩ 0,1 β‰  βˆ…

That is, the doubled slit tori that have a short horizontal saddle connection.

π›­πœ•

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Transversal for doubled slit tori

Consider the transversal for doubled slit tori

𝒳 = πœ•πœ—β„°|π›­πœ• ∩ 0,1 β‰  βˆ…

That is, the doubled slit tori that have a short horizontal saddle connection.

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Key: slope gaps = return times to 𝒳

  • First return time:

If πœ•πœ—π’³, when is β„Žπ‘£πœ•πœ—π’³?

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Key: slope gaps = return times to 𝒳

  • First return time:

If πœ•πœ—π’³, when is β„Žπ‘£πœ•πœ—π’³? Need a vector in π›­πœ• with β„Žπ‘£ 𝑦 𝑧 = 𝑦 𝑧 βˆ’ 𝑣𝑦 short and horizontal.

π›­πœ•

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Key: slope gaps = return times to 𝒳

  • First return time:

If πœ•πœ—π’³, when is β„Žπ‘£πœ•πœ—π’³? Need a vector in π›­πœ• with β„Žπ‘£ 𝑦 𝑧 = 𝑦 𝑧 βˆ’ 𝑣𝑦 short and horizontal.

  • This happens is when

𝑧 βˆ’ 𝑣𝑦 = 0 ⇔ 𝑣 = 𝑧 𝑦

π›­πœ•

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So the first return time is a slope

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So the first return time is a slope What about the second return time?

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Second return time = total time minus the first return time

Second return time π›­πœ•

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Second return time = total time minus the first return time Hence, second return time is a slope difference.

Second return time π›­πœ•

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Formalizing the key idea

Let 𝑆 denote the return time Let π‘ˆ denote the return map

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Formalizing the key idea

Let 𝑆 denote the return time 𝑆 πœ• = inf 𝑣 > | 0 β„Žπ‘£ πœ• ∈ 𝒳 Let π‘ˆ denote the return map

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Formalizing the key idea

Let 𝑆 denote the return time 𝑆 πœ• = inf 𝑣 > | 0 β„Žπ‘£ πœ• ∈ 𝒳 Let π‘ˆ denote the return map π‘ˆ πœ• = β„Žπ‘† πœ• πœ•

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SLIDE 113

Formalizing the key idea

Let 𝑆 denote the return time 𝑆 πœ• = inf 𝑣 > | 0 β„Žπ‘£ πœ• ∈ 𝒳 Let π‘ˆ denote the return map π‘ˆ πœ• = β„Žπ‘† πœ• πœ• ∈ 𝒳

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Formalizing the key idea

slope gaps = return times to 𝒳

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Formalizing the key idea

slope gaps = return times to 𝒳 𝑑𝑗+1 βˆ’ 𝑑𝑗 = 𝑆 π‘ˆπ‘— πœ•

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Slope gaps as a dynamical question

π»π‘π‘žπ‘‘π‘‚ π›­πœ• ∩ 𝐽 𝑂

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Slope gaps as a dynamical question

π»π‘π‘žπ‘‘π‘‚ π›­πœ• ∩ 𝐽 𝑂 = 1 𝑂 ෍

𝑗=0 π‘‚βˆ’1

πœ“ π‘†βˆ’1 𝐽 (π‘ˆπ‘— πœ• )

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SLIDE 118

Slope gaps as a dynamical question

π»π‘π‘žπ‘‘π‘‚ π›­πœ• ∩ 𝐽 𝑂 = 1 𝑂 ෍

𝑗=0 π‘‚βˆ’1

πœ“ π‘†βˆ’1 𝐽 (π‘ˆπ‘— πœ• ) β†’ 𝜈 πœ• | πœ—π’³ 𝑆 πœ• ∈ 𝐽

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Slope gaps as a dynamical question

π»π‘π‘žπ‘‘π‘‚ π›­πœ• ∩ 𝐽 𝑂 = 1 𝑂 ෍

𝑗=0 π‘‚βˆ’1

πœ“ π‘†βˆ’1 𝐽 (π‘ˆπ‘— πœ• ) β†’ 𝜈 πœ• | πœ—π’³ 𝑆 πœ• ∈ 𝐽 So next steps:

  • parametrize 𝒳
  • find return map in coordinates
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SLIDE 120

Part 2: Finding the return time

Return time = slope of the next vector to become short

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SLIDE 121

Part 2: Finding the return time

Return time = slope of the next vector to become short The rest of the talk we will

  • nly concern ourselves

with vectors of smallest positive slope

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SLIDE 122

Understanding saddle connections

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SLIDE 123

Understanding saddle connections

ΰ΅— β„‚ β„€2 , 1/2 1/2

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SLIDE 124

Understanding saddle connections

Two types of saddle connections

  • β„€2

ΰ΅— β„‚ β„€2 , 1/2 1/2

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SLIDE 125

Understanding saddle connections

Two types of saddle connections

  • β„€2
  • β„€2+ 1/2

1/2

ΰ΅— β„‚ β„€2 , 1/2 1/2

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SLIDE 126

ΰ΅— β„‚ 𝑕℀2 , 𝑀

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SLIDE 127

ΰ΅— β„‚ 𝑕℀2 , 𝑀 Two types of saddle connections

  • 𝑕℀2
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SLIDE 128

ΰ΅— β„‚ 𝑕℀2 , 𝑀 Two types of saddle connections

  • 𝑕℀2

Understood by torus results

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SLIDE 129

ΰ΅— β„‚ 𝑕℀2 , 𝑀 Two types of saddle connections

  • 𝑕℀2

Understood by torus results

  • 𝑕℀2 + 𝑀
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SLIDE 130

ΰ΅— β„‚ 𝑕℀2 , 𝑀 Two types of saddle connections

  • 𝑕℀2

Understood by torus results

  • 𝑕℀2 + 𝑀

Defines an affine lattice!

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SLIDE 131

Parameterizing affine lattices

Data needed for an affine lattice 𝛭 = 𝑕℀2 + 𝑀 is

  • lattice 𝑕 ∈ 𝑇𝑀 2, ℝ
  • vector 𝑀 ∈

Ξ€ β„‚ 𝑕 β„€2 𝛭 = 𝑕℀2 + 𝑀.

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SLIDE 132

Given an affine lattice 𝛭 = 𝑕℀2 + 𝑀, what is the short vector of smallest slope? 𝛭 = 𝑕℀2 + 𝑀.

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SLIDE 133

A special case

Consider the affine lattices of the form 𝛭 = 1 𝑐 1 β„€2 + 𝛽 0 . What are the vectors of smallest slope?

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SLIDE 134

𝛭 = 1 𝑐 1 β„€2 + 𝛽

At every height, can have at most

  • ne vector in a

unit length interval.

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SLIDE 135

Strategy for 𝛭 = 1 𝑐 1 β„€2 + 𝛽

So to find vector of smallest non-zero slope

  • Consider the affine vector

𝛽 0 .

  • Use structure of the lattice

and track how slope changes

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SLIDE 136

Strategy for 𝛭 = 1 𝑐 1 β„€2 + 𝛽

So to find vector of smallest non-zero slope

  • Consider the affine vector

𝛽 0 .

  • Use structure of the lattice

and track how slope changes

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SLIDE 137

Short vectors of 𝛭 = 1 𝑐 1 β„€2 + 𝛽

The next vector to become short ࡞ 𝛽 0 + second basis vector , 𝑗𝑔 𝑐 + 𝛽 < 1 𝛽 0 βˆ’ first basis vector + (many) second basis , 𝑗𝑔 𝑐 + 𝛽 > 1

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SLIDE 138

Short vectors of 𝛭 = 1 𝑐 1 β„€2 + 𝛽

The next vector to become short 𝑐 + 𝛽 1 , 𝑗𝑔 𝑐 + 𝛽 < 1 π‘˜π‘ + 𝛽 βˆ’ 1 j , 𝑗𝑔 𝑐 + 𝛽 > 1 where π‘˜ =

2βˆ’π›½ 𝑐

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SLIDE 139

Elements of the proof

  • This idea (with some modifications) is used to find

holonomy vectors of doubled slit tori of smallest slope

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SLIDE 140

Elements of the proof

  • This idea (with some modifications) is used to find

holonomy vectors of doubled slit tori of smallest slope

  • These are the return times to the transversal
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SLIDE 141

Elements of the proof

  • This idea (with some modifications) is used to find

holonomy vectors of doubled slit tori of smallest slope

  • These are the return times to the transversal
  • This answer answers the gap distribution question for

doubled slit tori

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SLIDE 142

Special thanks to:

  • Dr. Jayadev Athreya (My advisor)
  • West Coast Dynamics Seminar