A mathematical preprint identifies a strict gap below 1 in the entropy spectrum of every closed hyperbolic surface: the upper limit of the entropies of its proper subsurfaces is less than 1. The size of that gap depends on the surface’s geometry, so the result does not give one universal numerical value.
An entropy spectrum here means a multiset—the values are counted with their repetitions—of the entropies of connected subsurfaces. The analysis covers connected, orientable, finite-type hyperbolic surfaces whose boundaries consist of simple closed geodesics or cusps. In this setting, entropy is tied to how quickly counts of closed and non-filling geodesics grow as the allowed length increases.
An ordered spectrum
The spectrum has a precise order. The paper rules out any nonincreasing sequence of values indexed by distinct subsurfaces; in its terms, the spectrum is reverse well-ordered and has bounded multiplicities. Put more simply, distinct subsurfaces cannot generate an endless nonincreasing run of entropy values, and repeated values remain controlled.
Accumulation points—the values that can be approached by other values in the spectrum—are characterized exactly. They are the entropies of connected subsurfaces whose complements have curve complexity at least 1, using the paper’s measure of the curve structure left in the complement.
The paper also gives a geometric description of the largest proper entropy. It is the maximum entropy among the complements of non-separating simple closed geodesics—geodesics whose removal does not split the surface—and any subsurface attaining that value is one of those complements.
Counting geodesics
To obtain its boundary-surface estimate, the construction uses ideal triangles separated by strips and then collapses the strips, so boundary components become cusps on a complete finite-area surface.
For a surface with positive boundary length B and boundary width W, the number of closed geodesics of length at most L has an upper bound of C e^{λL}. The constants C and λ are explicit functions of B and W; λ is strictly below 1, and the surface entropy h_X is no greater than λ.
For a closed hyperbolic surface of genus g with systole at least ε—the systole is the length of the shortest non-shrinkable closed geodesic—the number of non-filling geodesics of length at most L is bounded by A e^{aL}. Here A and a depend on g and ε, and a<1. The proof gets this by finding a simple closed geodesic in the complement of each non-filling curve, then combining simple-geodesic counts with the boundary-surface estimate.
Results with conditions
The results also include a rigidity statement: the marked version of the entropy spectrum determines a closed hyperbolic surface up to isometry—that is, up to a distance-preserving identification. In a specified attachment construction, uniformly bounded widths and systoles tending to infinity imply that the entropy of the glued surfaces converges to the entropy of the fixed piece.
These conclusions remain tied to the stated conditions: the boundary bound depends on boundary length and width, while the non-filling bound is stated for closed surfaces with genus and systole restrictions. The document is arXiv:2608.20143v1, dated 20 Aug 2026.
Paper data and sources
Original title: The entropy spectrum of hyperbolic surfaces
Authors: Ara Basmajian, Hugo Parlier
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-20
DOI: Not available
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