Topological entropy of saturated sets for Lorenz attractors
Abstract
We study the topological entropy of saturated sets for continuous flows, with emphasis on singular hyperbolic dynamics. For a Cr-generic (r ≥ 2) vector field X carrying a geometric Lorenz attractor Λ, we prove that for every nonempty compact connected set 𝒦 ⊆ ℳ(Φ,Λ) and every open set U ⊆ M with U ∩ Λ≠ ∅, htop(Φ,G𝒦(Φ) ∩ Trans(Φ,Λ) ∩ U)=infμ∈ 𝒦 hμ(ϕ1), where Φ=(ϕt)t ∈ ℝ is the flow induced by X and Trans(Φ,Λ) denotes the set of points whose positive orbits are dense in Λ. The same formula holds C1-generically for singular hyperbolic attractors. In contrast with the uniformly hyperbolic case, we also prove that, for vector fields in a dense subset of 𝒳r(M3), geometric Lorenz attractors admit a positive-entropy invariant measure with no generic points. An analogous conclusion holds for vector fields in a dense subset of 𝒳1(M) whose non-trivial singular hyperbolic attractors contain singularities and satisfy dim Ecu=2. These examples show that the discrete-time saturated entropy theory does not extend directly to singular flows.
Related
- Full Topological entropy of Xiong chaotic sets beyond the specification property
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- Dense times-b orbits from times-a invariant sets and measures, in zero entropy
BibTeX
@misc{huang2026topological,
title = {{Topological entropy of saturated sets for Lorenz attractors}},
author = {Zheng Huang and Xueting Tian and Xiaodong Wang and Xutong Zhao},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.18148},
eprint = {2610.18148},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.18148},
}
RIS
TY - UNPB
TI - Topological entropy of saturated sets for Lorenz attractors
AU - Zheng Huang
AU - Xueting Tian
AU - Xiaodong Wang
AU - Xutong Zhao
PY - 2026
DA - 2026/10/13
PB - arXiv
DO - 10.48550/arxiv.2610.18148
UR - https://arxiv.org/abs/2610.18148
AB - We study the topological entropy of saturated sets for continuous flows, with emphasis on singular hyperbolic dynamics. For a C^r-generic (r ≥ 2) vector field X carrying a geometric Lorenz attractor Λ, we prove that for every nonempty compact connected set 𝒦 ⊆ ℳ(Φ,Λ) and every open set U ⊆ M with U ∩ Λ≠ ∅, h_top(Φ,G_𝒦(Φ) ∩ Trans(Φ,Λ) ∩ U)=inf_μ∈ 𝒦 h_μ(ϕ_1), where Φ=(ϕ_t)_t ∈ ℝ is the flow induced by X and Trans(Φ,Λ) denotes the set of points whose positive orbits are dense in Λ. The same formula holds C^1-generically for singular hyperbolic attractors. In contrast with the uniformly hyperbolic case, we also prove that, for vector fields in a dense subset of 𝒳^r(M^3), geometric Lorenz attractors admit a positive-entropy invariant measure with no generic points. An analogous conclusion holds for vector fields in a dense subset of 𝒳^1(M) whose non-trivial singular hyperbolic attractors contain singularities and satisfy dim E^cu=2. These examples show that the discrete-time saturated entropy theory does not extend directly to singular flows.
ER -