Almost sure orbits closeness
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866911211104239616 |
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| author | Kirsebom, Maxim Kunde, Philipp Persson, Tomas Todd, Mike |
| author_facet | Kirsebom, Maxim Kunde, Philipp Persson, Tomas Todd, Mike |
| contents | We consider the minimal distance between orbits of measure
preserving dynamical systems. In the spirit of dynamical
shrinking target problems we identify distance rates for which
almost sure asymptotic closeness properties can be
ensured. More precisely, we consider the set $E_n$ of pairs of
points whose orbits up to time $n$ have minimal distance to
each other less than the threshold $r_n$. We obtain bounds on
the sequence $(r_n)_n$ to guarantee that $\limsup_{n}E_n$ and
$\liminf_{n} E_n$ are sets of measure 0 or 1. Results for the
measure 0 case are obtained in broad generality while the
measure one case requires assumptions of exponential mixing for
at least one of the systems. We also consider the analogous
question of the minimal distance of points within a single
orbit of one dimensional exponentially mixing dynamical systems. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_13277 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Almost sure orbits closeness Kirsebom, Maxim Kunde, Philipp Persson, Tomas Todd, Mike Dynamical Systems 37B20, 37A05, 37D05, 37E05 We consider the minimal distance between orbits of measure preserving dynamical systems. In the spirit of dynamical shrinking target problems we identify distance rates for which almost sure asymptotic closeness properties can be ensured. More precisely, we consider the set $E_n$ of pairs of points whose orbits up to time $n$ have minimal distance to each other less than the threshold $r_n$. We obtain bounds on the sequence $(r_n)_n$ to guarantee that $\limsup_{n}E_n$ and $\liminf_{n} E_n$ are sets of measure 0 or 1. Results for the measure 0 case are obtained in broad generality while the measure one case requires assumptions of exponential mixing for at least one of the systems. We also consider the analogous question of the minimal distance of points within a single orbit of one dimensional exponentially mixing dynamical systems. |
| title | Almost sure orbits closeness |
| topic | Dynamical Systems 37B20, 37A05, 37D05, 37E05 |
| url | https://arxiv.org/abs/2510.13277 |