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Temporal Metric Spaces and Temporal Iterated Function Systems

  • Amal P. S.
  • Vinod Kumar P. B.
  • Ramkumar P. B
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Abstract

We introduce temporal metric spaces, a framework for formalizing metric and topological structures on continuously evolving domains. Rather than defining a single metric on a fixed space, we construct a family of interval-dependent metrics on an associated trajectory space, enabling distances to be measured between continuous trajectories over finite time intervals. We establish the fundamental topology of the trajectory space, prove that it inherits completeness from the underlying metric space, and develop the corresponding temporal hyperspace together with its completeness. Finally, we develop a theory of iterated function systems acting on these evolving spaces, proving that their attractors evolve coherently with the underlying dynamics and remain stable even when the evolving geometry causes the generating mappings to lose their classical contraction property.

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Cite

BibTeXRIS
BibTeX
@misc{p2026temporal,
  title = {{Temporal Metric Spaces and Temporal Iterated Function Systems}},
  author = {Amal P. S. and Vinod Kumar P. B. and Ramkumar P. B},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.05805},
  eprint = {2610.05805},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.05805},
}
RIS
TY  - UNPB
TI  - Temporal Metric Spaces and Temporal Iterated Function Systems
AU  - Amal P. S.
AU  - Vinod Kumar P. B.
AU  - Ramkumar P. B
PY  - 2026
DA  - 2026/10/05
PB  - arXiv
DO  - 10.48550/arxiv.2610.05805
UR  - https://arxiv.org/abs/2610.05805
AB  - We introduce temporal metric spaces, a framework for formalizing metric and topological structures on continuously evolving domains. Rather than defining a single metric on a fixed space, we construct a family of interval-dependent metrics on an associated trajectory space, enabling distances to be measured between continuous trajectories over finite time intervals. We establish the fundamental topology of the trajectory space, prove that it inherits completeness from the underlying metric space, and develop the corresponding temporal hyperspace together with its completeness. Finally, we develop a theory of iterated function systems acting on these evolving spaces, proving that their attractors evolve coherently with the underlying dynamics and remain stable even when the evolving geometry causes the generating mappings to lose their classical contraction property.
ER  -