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Counting and entropy for hyperbolic surface amalgams

Hugo Parlier, Yandi Wu

Abstract

This paper is about closed hyperbolic surface amalgams with a focus on the growth of the number of closed geodesics. As in the case of surfaces, we show that topological and volume entropies coincide, but we show stark differences in how they behave according to geometric data with upper and lower bounds on the number of closed geodesics which depend on the length of the systole and the length of the pasting curves. In particular, we show that the entropy can increase exponentially in terms of the pasting length in the absence of a lower bound on the systole.

Counting and entropy for hyperbolic surface amalgams

Abstract

This paper is about closed hyperbolic surface amalgams with a focus on the growth of the number of closed geodesics. As in the case of surfaces, we show that topological and volume entropies coincide, but we show stark differences in how they behave according to geometric data with upper and lower bounds on the number of closed geodesics which depend on the length of the systole and the length of the pasting curves. In particular, we show that the entropy can increase exponentially in terms of the pasting length in the absence of a lower bound on the systole.
Paper Structure (9 sections, 23 theorems, 110 equations, 11 figures)

This paper contains 9 sections, 23 theorems, 110 equations, 11 figures.

Table of Contents

  1. Introduction
  2. Preliminaries and technical lemmas
  3. Hyperbolic geometry
  4. Surface amalgams
  5. Entropies and the count of geodesics
  6. Counting Geodesics
  7. Upper Bound
  8. Long gluing curves on fixed surfaces
  9. Short systoles and high entropy

Key Result

Theorem 1.1

Let $X$ be proper surface amalgam of area $A$, of total gluing length $B$ and let $r_0$ be as above. Then As a consequence, $h(X)$, the entropy of $X$, satisfies

Figures (11)

  • Figure 1: The complementary regions of $\gamma \cap D$ must either be quadrilaterals or half disks if $D$ has radius less than or equal to $\frac{\log(3)}{2}$.
  • Figure 2:
  • Figure 3: An example of a proper hyperbolic surface amalgam with four chambers.
  • Figure 4: There are more topological types for collars of short curves in surface amalgams.
  • Figure 5: If two short geodesics intersect at a gluing curve, the images of their lifts may project to the same bi-infinite closed geodesic, so the proof for \ref{['lemma:transverse']} will not work.
  • ...and 6 more figures

Theorems & Definitions (48)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Lemma 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3: c.f. Lemma A.1, bp
  • Corollary 2.4
  • proof
  • Definition 2.5: Hyperbolic surface amalgam, c.f. Definition 2.3 of lafont
  • ...and 38 more