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PreprintCC BY 4.0arXiv

Physical measures for partially hyperbolic systems with u-definite subbundles

  • Yongluo Cao
  • Zeya Mi
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Abstract

In this paper, we study C1+α diffeomorphisms of a compact Riemannian manifold M with an attracting set Λ admitting a partially hyperbolic splitting TΛM = Eu⊕ C1⊕⋯⊕ Ck⊕ Es. We assume that, for every ergodic Gibbs u-state supported on Λ, the Lyapunov exponents along each Ci are either all positive or all negative. We prove that Λ supports finitely many ergodic hyperbolic SRB measures whose basins cover Lebesgue almost every point of its topological basin. Our approach relies on an empirical basin-covering theorem for attracting sets with dominated splittings. Its proof hinges on a new notion of sliding suffix times, which allows us to synchronize expansion and contraction along a common empirical subsequence.

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Cite

BibTeXRIS
BibTeX
@misc{cao2026physical,
  title = {{Physical measures for partially hyperbolic systems with $u$-definite subbundles}},
  author = {Yongluo Cao and Zeya Mi},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.24743},
  eprint = {2610.24743},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.24743},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Physical measures for partially hyperbolic systems with u-definite subbundles
AU  - Yongluo Cao
AU  - Zeya Mi
PY  - 2026
DA  - 2026/10/17
PB  - arXiv
DO  - 10.48550/arxiv.2610.24743
UR  - https://arxiv.org/abs/2610.24743
AB  - In this paper, we study C^1+α diffeomorphisms of a compact Riemannian manifold M with an attracting set Λ admitting a partially hyperbolic splitting T_ΛM = E^u⊕ C_1⊕⋯⊕ C_k⊕ E^s. We assume that, for every ergodic Gibbs u-state supported on Λ, the Lyapunov exponents along each C_i are either all positive or all negative. We prove that Λ supports finitely many ergodic hyperbolic SRB measures whose basins cover Lebesgue almost every point of its topological basin. Our approach relies on an empirical basin-covering theorem for attracting sets with dominated splittings. Its proof hinges on a new notion of sliding suffix times, which allows us to synchronize expansion and contraction along a common empirical subsequence.
ER  -