Gromov-Hausdorff distance and stability of dynamical systems

@misc{abdullaev2026gromovhausdorff,
  title = {{Gromov-Hausdorff distance and stability of dynamical systems}},
  author = {Emin Abdullaev},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2608.02906},
  eprint = {2608.02906},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2608.02906},
}

TY  - UNPB
TI  - Gromov-Hausdorff distance and stability of dynamical systems
AU  - Emin Abdullaev
PY  - 2026
DA  - 2026/10/02
PB  - arXiv
DO  - 10.48550/arxiv.2608.02906
UR  - https://arxiv.org/abs/2608.02906
AB  - We introduce the notion of an F-space - a set equipped with a family of generalized pseudometrics and marked points, and construct the categorical Gromov-Hausdorff distance for F-spaces. Based on this, we propose a new definition of the distance between dynamical systems, understood in a broad sense as parameterized families of maps. The main focus is on local dynamics: we prove the preservation of Lyapunov stability and asymptotic stability under limit transitions with respect to the new distance (in the latter case, assuming a common radius of attraction). It is established that the class of stable systems is a closed and nowhere dense subset in the space of compact dynamical systems. Next, we investigate the properties of the Hausdorff map, which induces an F-space structure on the family of subsets. In the final part, we introduce another modified version of the Gromov-Hausdorff distance for dynamical systems and use it to calculate the exact distance between torus translations for a non-resonant vector.
ER  -