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Natural Expansiveness and Equilibrium States for Semiflows

  • Nelda Tamblay
  • Juan Francisco Peña
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Abstract

We introduce natural expansiveness, a notion of expansiveness for non-invertible dynamical systems which is defined by using the natural extension. We show that every expansive homeomorphism or flow, every positive expansive endomorphism and every positive expansive semiflow without end points is naturally expansive. Moreover, we prove that the suspension of a naturally expansive endomorphism is naturally expansive. This gives examples of naturally expansive semiflows which are neither positive expansive nor restrictions of expansive flows. Then we extend natural expansiveness and the topological pressure to semiflows which are not necessarily continuous, by using a pseudosemimetric dρ. We introduce the notion of dρ-conjugacy, and we prove that every regular impulsive semiflow is dρ-conjugate to a continuous semiflow. As an application, we obtain the existence and uniqueness of equilibrium states for regular impulsive semiflows.

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Cite

BibTeXRIS
BibTeX
@misc{tamblay2026natural,
  title = {{Natural Expansiveness and Equilibrium States for Semiflows}},
  author = {Nelda Tamblay and Juan Francisco Peña},
  howpublished = {arXiv},
  year = {2026},
  month = nov,
  doi = {10.48550/arxiv.2611.06565},
  eprint = {2611.06565},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2611.06565},
}
RIS
TY  - UNPB
TI  - Natural Expansiveness and Equilibrium States for Semiflows
AU  - Nelda Tamblay
AU  - Juan Francisco Peña
PY  - 2026
DA  - 2026/11/04
PB  - arXiv
DO  - 10.48550/arxiv.2611.06565
UR  - https://arxiv.org/abs/2611.06565
AB  - We introduce natural expansiveness, a notion of expansiveness for non-invertible dynamical systems which is defined by using the natural extension. We show that every expansive homeomorphism or flow, every positive expansive endomorphism and every positive expansive semiflow without end points is naturally expansive. Moreover, we prove that the suspension of a naturally expansive endomorphism is naturally expansive. This gives examples of naturally expansive semiflows which are neither positive expansive nor restrictions of expansive flows. Then we extend natural expansiveness and the topological pressure to semiflows which are not necessarily continuous, by using a pseudosemimetric d_ρ. We introduce the notion of d_ρ-conjugacy, and we prove that every regular impulsive semiflow is d_ρ-conjugate to a continuous semiflow. As an application, we obtain the existence and uniqueness of equilibrium states for regular impulsive semiflows.
ER  -