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Pair correlation statistics for dynamical systems

  • Simon Baker
  • Mike Todd
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Abstract

We study the pair correlation statistics of orbits generated by maps on the interval. We show that under suitable mixing and multifractal assumptions, the pair correlation statistics of an orbit will almost surely exhibit the same asymptotic behaviour as a suitable sequence of i.i.d. random variables. We will also show that, under suitable hypotheses, the pair correlation statistics defined by two orbits will almost surely exhibit the same behaviour as two suitable sequences of i.i.d. random variables. Specific dynamical systems to which our results apply to include Gibbs-Markov maps and the Gauss map. We also give an example of a slowly mixing system for which the pair correlation statistics of an orbit almost surely behave distinctly to an i.i.d. sequence.

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Cite

BibTeXRIS
BibTeX
@misc{baker2026pair,
  title = {{Pair correlation statistics for dynamical systems}},
  author = {Simon Baker and Mike Todd},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2606.17880},
  eprint = {2606.17880},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2606.17880},
}
RIS
TY  - UNPB
TI  - Pair correlation statistics for dynamical systems
AU  - Simon Baker
AU  - Mike Todd
PY  - 2026
DA  - 2026/09/03
PB  - arXiv
DO  - 10.48550/arxiv.2606.17880
UR  - https://arxiv.org/abs/2606.17880
AB  - We study the pair correlation statistics of orbits generated by maps on the interval. We show that under suitable mixing and multifractal assumptions, the pair correlation statistics of an orbit will almost surely exhibit the same asymptotic behaviour as a suitable sequence of i.i.d. random variables. We will also show that, under suitable hypotheses, the pair correlation statistics defined by two orbits will almost surely exhibit the same behaviour as two suitable sequences of i.i.d. random variables. Specific dynamical systems to which our results apply to include Gibbs-Markov maps and the Gauss map. We also give an example of a slowly mixing system for which the pair correlation statistics of an orbit almost surely behave distinctly to an i.i.d. sequence.
ER  -