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arxiv: 2604.05174 · v1 · submitted 2026-04-06 · 🧮 math.DS · math.DG· math.GT

Entropy and self-intersection number of geodesic currents on compact hyperbolic surfaces

Tina Torkaman This is my paper

Pith reviewed 2026-05-10 18:52 UTC · model grok-4.3

classification 🧮 math.DS math.DGmath.GT
keywords geodesic currentsself-intersection numbermeasure-theoretic entropyhyperbolic surfacesgeodesic flow
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The pith

The entropy of an ergodic geodesic current is bounded above by its self-intersection number and the systole of the surface.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper establishes a quantitative upper bound for the measure-theoretic entropy h_X(C) of an ergodic geodesic current C on a compact hyperbolic surface X. The bound is expressed in terms of the self-intersection number i(C,C) and the systole of X. If true, this shows that geodesic currents with small self-intersection must have small entropy. A sympathetic reader would care because it provides a way to control the dynamical entropy using a geometric quantity in the context of the geodesic flow on hyperbolic surfaces.

Core claim

We establish a quantitative upper bound on h_X(C) in terms of its self-intersection number i(C,C) and the systole of X. In particular, we show that small self-intersection number forces small entropy for ergodic geodesic currents C.

What carries the argument

The self-intersection number i(C,C) of the geodesic current C, which is used to bound the entropy h_X(C) of the geodesic flow.

If this is right

  • Ergodic geodesic currents with zero self-intersection have zero entropy.
  • Entropy increases as self-intersections increase, modulated by the systole length.
  • The bound gives a way to estimate entropy without directly computing the measure-theoretic entropy.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • This bound could be used to study the space of geodesic currents with bounded entropy.
  • Extensions might include non-compact surfaces or non-ergodic currents by decomposition.
  • Computations on specific surfaces like the once-punctured torus could test the sharpness of the bound.

Load-bearing premise

The geodesic current C must be ergodic for the bound to hold.

What would settle it

An explicit example of an ergodic geodesic current on a compact hyperbolic surface where the entropy exceeds the upper bound given by the self-intersection number and systole would disprove the claim.

Figures

Figures reproduced from arXiv: 2604.05174 by Tina Torkaman.

Figure 1
Figure 1. Figure 1: The closed curve φ(γ) does not have intersection of this type Question. Can the ergodicity assumption in Theorem 1.1 be removed? Specifically, if geodesic currents {Cn} satisfy i(Cn, Cn) → 0 as n → ∞, does it follow that hX(Cn) → 0? 4 [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: In this hexagon decompisition of a genus [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: The subarc α has two intersection points q1, q2 on an edge ei inside a hexagon obtained from the seams of the pair of pants, this arc has length ≥ sys(X)/2. Combining the two cases, we obtain ℓcom(γ) ≤ 7 i(γ, p) + 12 ℓX(γ) sys(X) . 9 [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Different types of intersection between two arcs [PITH_FULL_IMAGE:figures/full_fig_p010_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: ). We define the proper ordering of the points p1, . . . , pn on ei to be the one that is compatible with the order of f +(p1), . . . , f+(pn) along this boundary interval of ∂D. Equivalently, the permutation of the intersection points on ei that agrees with the order of their forward endpoints on ∂D is defined as the proper ordering; see [PITH_FULL_IMAGE:figures/full_fig_p012_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: We determine the relative positions of points on each edge according to [PITH_FULL_IMAGE:figures/full_fig_p013_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Point q divides the points on e1 into two blocks By iterating this procedure four times, we obtain the desired result for a hexagon. Equivalently, we first count the number of words formed using two symbols, e2 and b, where the symbol b means we have an arc from e1 to one of the remaining four edges e3, e4, e5, e6. Once such words are fixed, determining how the positions labeled by b are assigned to the sp… view at source ↗
Figure 8
Figure 8. Figure 8: A hyperbolic 4-gon Proof. The proof is implied from Equation (2, 3, 2) in [Bus]. Let f[0,t](v) be the geodesic arc of length t in T1(X) starting at v ∈ T1(X). Proposition 4.5. Let c0 > 0 be a sufficiently small constant depending on X. Let C be an ergodic geodesic current. Then for C-almost every v ∈ T1(X), and for any sequence tn → ∞ satisfying d [PITH_FULL_IMAGE:figures/full_fig_p021_8.png] view at source ↗
read the original abstract

Let $X$ be a compact hyperbolic surface of genus $g$, and $C$ a geodesic current on $X$. Denote by $h_X(C)$ the measure-theoretic entropy of $C$ with respect to the geodesic flow. Assume that $C$ is ergodic. In this paper, we establish a quantitative upper bound on $h_X(C)$ in terms of its self-intersection number $i(C,C)$ and the systole of $X$. In particular, we show that small self-intersection number forces small entropy.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 1 minor

Summary. Let X be a compact hyperbolic surface of genus g and C an ergodic geodesic current on X. The paper establishes a quantitative upper bound on the measure-theoretic entropy h_X(C) of C with respect to the geodesic flow, expressed in terms of the self-intersection number i(C,C) and the systole of X. The central conclusion is that small self-intersection number forces small entropy.

Significance. If the bound holds, the result supplies a new relation between a dynamical quantity (entropy of the geodesic flow) and a geometric invariant (self-intersection) for geodesic currents. The explicit dependence on the systole is geometrically natural, as it furnishes a uniform lower bound on injectivity radius that controls local expansion. This could prove useful for rigidity or classification questions involving low-entropy currents on hyperbolic surfaces.

minor comments (1)
  1. Abstract: the explicit form of the upper bound is not displayed, which would allow readers to assess its sharpness and dependence on the systole immediately.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our manuscript, the recognition of its significance in relating entropy of geodesic currents to self-intersection number and systole, and the recommendation for minor revision. No major comments were provided in the report.

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper establishes a mathematical upper bound relating measure-theoretic entropy h_X(C) of an ergodic geodesic current to its self-intersection number i(C,C) and the systole of the compact hyperbolic surface X. No self-definitional relations appear (entropy is not defined via self-intersection), no parameters are fitted to data and then relabeled as predictions, and no load-bearing steps reduce to self-citations or ansatzes imported from prior work by the same author. The ergodicity assumption is standard for defining the entropy of the geodesic flow, and the dependence on systole is a natural geometric control on local expansion. The claimed result is a theorem proved from first principles in hyperbolic geometry and dynamical systems, independent of its own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review provides no details on free parameters, axioms, or invented entities used in the proof.

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Reference graph

Works this paper leans on

4 extracted references · 4 canonical work pages

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    [KU] Svetlana Katok and Ilie Ugarcovici. Symbolic dynamics for the modular sur- face and beyond.Bulletin of the American Mathematical Society44(2007), 87–132. [MT] Curtis T McMullen and Tina Torkaman. Teichmüller theory via random sim- ple closed curves.arXiv preprint arXiv:2512.14101(2025). [Sap1] Jenya Sapir. Bounds on the number of non-simple closed ge...

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    Train tracks, entropy, and the halo of a measured lamination.arXiv preprint arXiv:2105.00370(2021)

    [TZ] Tina Torkaman and Yongquan Zhang. Train tracks, entropy, and the halo of a measured lamination.arXiv preprint arXiv:2105.00370(2021). 27

    work page arXiv 2021