Conley index theory for asymptotically autonomous systems and UV-to-IR connections
Abstract
Many non-equilibrium systems, such as expanding plasmas, evolve by equations that depend explicitly on time and interpolate between autonomous early-time (UV) and late-time (IR) limits. After time is compactified, fixed points lie on invariant boundary fibers, where the Conley index is one-sided and blind to the UV. For time dependence that admits a C1 compactification, we reflect the extended phase space across these fibers. Both sheets of the double run from the UV to the IR, and the UV index reappears, suspended by one degree. Extending one sheet slightly beyond both fibers gives an explicit UV-to-IR transport map between the Conley indices of the limit systems, and its non-vanishing forces connecting solutions. If one block with a fixed strict entrance/exit decomposition serves every frozen system, this map is an isomorphism at every admissible rate. If the block also has nonzero relative homology, the isolated invariant sets of the two limits are connected at every rate. In the Gubser flow of Israel-Stewart theory, the separatrix between two classes of initial data is a transversal connection at every finite relaxation scale c, with splitting Δ(c)∝1/c. In the Bjorken flow at the lowest moment truncation, we prove existence and uniqueness of the attractor solution without regularization. For the three-dimensional Gubser system we construct the global Morse decomposition. In the one-loop flow of the (4+ε)-dimensional O(N) model with cubic anisotropy, a shift of N through N=4 has vanishing transport map, but a driven connection and a nonzero connecting homomorphism detect it at every rate.
Related
- Asymptotically Autonomous Maps: Heteroclinic connections, Splitting and Phase-induced tipping
- A Chern-Weil theory for Lie groupoids with Cartan connections
- GKZ Systems and Gauss-Manin Connections of Families of Projective Hypersurfaces
- Eliminating the Higgs field and perturbing Yang-Mills connections in cone Yang-Mills theory
- Yang-Mills theory for multiplicative Ehresmann connections
BibTeX
@misc{behtash2026conley,
title = {{Conley index theory for asymptotically autonomous systems and UV-to-IR connections}},
author = {Alireza Behtash and Syo Kamata and Mauricio M. Martinez},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.08591},
eprint = {2611.08591},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.08591},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Conley index theory for asymptotically autonomous systems and UV-to-IR connections
AU - Alireza Behtash
AU - Syo Kamata
AU - Mauricio M. Martinez
PY - 2026
DA - 2026/11/05
PB - arXiv
DO - 10.48550/arxiv.2611.08591
UR - https://arxiv.org/abs/2611.08591
AB - Many non-equilibrium systems, such as expanding plasmas, evolve by equations that depend explicitly on time and interpolate between autonomous early-time (UV) and late-time (IR) limits. After time is compactified, fixed points lie on invariant boundary fibers, where the Conley index is one-sided and blind to the UV. For time dependence that admits a C^1 compactification, we reflect the extended phase space across these fibers. Both sheets of the double run from the UV to the IR, and the UV index reappears, suspended by one degree. Extending one sheet slightly beyond both fibers gives an explicit UV-to-IR transport map between the Conley indices of the limit systems, and its non-vanishing forces connecting solutions. If one block with a fixed strict entrance/exit decomposition serves every frozen system, this map is an isomorphism at every admissible rate. If the block also has nonzero relative homology, the isolated invariant sets of the two limits are connected at every rate. In the Gubser flow of Israel-Stewart theory, the separatrix between two classes of initial data is a transversal connection at every finite relaxation scale c, with splitting Δ(c)∝1/c. In the Bjorken flow at the lowest moment truncation, we prove existence and uniqueness of the attractor solution without regularization. For the three-dimensional Gubser system we construct the global Morse decomposition. In the one-loop flow of the (4+ε)-dimensional O(N) model with cubic anisotropy, a shift of N through N=4 has vanishing transport map, but a driven connection and a nonzero connecting homomorphism detect it at every rate.
ER -