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Geodesic tracking and the shape of ergodic rotation sets

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Alejo García-Sassi, Pierre-Antoine Guihéneuf, Pablo Lessa

Dynamical Systems · Submitted 11 Dec 2023

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim We prove a structure theorem for ergodic homological rotation sets of homeomorphisms isotopic to the identity on a closed orientable hyperbolic surface: this set is made of a finite number of pieces that are either one-dimensional or almost convex.

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arXiv ID2312.06249 (version 3)
Pith IDGG6DSKUU, Pith's permanent record of this paper and its review
CitationAlejo Garc\'ia-Sassi, Pablo Lessa, Pierre-Antoine Guih\'eneuf. "Geodesic tracking and the shape of ergodic rotation sets." arXiv:2312.06249 (2023). https://pith.science/paper/GG6DSKUU
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Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 3,212 input tokens.

What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.20 · human participants 0.01 · controlled comparison 0.01 · randomized assignment 0.01 · states quantitative result 0.14 · compares with prior results 0.06 · mentions code or data availability 0.01.

The abstract, sentence by sentence

arXiv listing abstract v3. Results and the main claim are underlined; .

We prove a structure theorem for ergodic homological rotation sets of homeomorphisms isotopic to the identity on a closed orientable hyperbolic surface: this set is made of a finite number of pieces that are either one-dimensional or almost convex. The latter ones give birth to horseshoes; in the case of a zero-entropy homeomorphism we show that there exists a geodesic lamination containing the directions in which generic orbits with respect to ergodic invariant probabilities turn around the surface under iterations of the homeomorphism. The proof is based on the idea of $\textit{geodesic tracking}$ of orbits that are typical for some invariant measure by geodesics on the surface, that allows to get links between the dynamics of such points and the one of the geodesic flow on some invariant subset of the unit tangent bundle of the surface.

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#sentencerolethe authors' own workstates the main conclusionstrength
s0 We prove a structure theorem for ergodic homological rotation sets of homeomorphisms isotopic to the identity on a closed orientable hyperbolic surface: this set is made of a finite number of pieces that are either one-dimensional or almost convex. result 0.93 0.95 0.90 asserted 1.00
s1 The latter ones give birth to horseshoes; in the case of a zero-entropy homeomorphism we show that there exists a geodesic lamination containing the directions in which generic orbits with respect to ergodic invariant probabilities turn around the surface under iterations of the homeomorphism. result 0.90 0.91 0.16 asserted 1.00
s2 The proof is based on the idea of $\textit{geodesic tracking}$ of orbits that are typical for some invariant measure by geodesics on the surface, that allows to get links between the dynamics of such points and the one of the geodesic flow on some invariant subset of the unit tangent bundle of the surface. method 1.00 0.87 0.05 asserted 0.99

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 1.00.

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Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

The paper
Original abstract

We prove a structure theorem for ergodic homological rotation sets of homeomorphisms isotopic to the identity on a closed orientable hyperbolic surface: this set is made of a finite number of pieces that are either one-dimensional or almost convex. The latter ones give birth to horseshoes; in the case of a zero-entropy homeomorphism we show that there exists a geodesic lamination containing the directions in which generic orbits with respect to ergodic invariant probabilities turn around the surface under iterations of the homeomorphism. The proof is based on the idea of $\textit{geodesic tracking}$ of orbits that are typical for some invariant measure by geodesics on the surface, that allows to get links between the dynamics of such points and the one of the geodesic flow on some invariant subset of the unit tangent bundle of the surface.

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