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Almost sure orbits closeness

Maxim Kirsebom, Philipp Kunde, Tomas Persson, Mike Todd

Abstract

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.

Almost sure orbits closeness

Abstract

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 of pairs of points whose orbits up to time have minimal distance to each other less than the threshold . We obtain bounds on the sequence to guarantee that and 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.
Paper Structure (9 sections, 13 theorems, 138 equations)

This paper contains 9 sections, 13 theorems, 138 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Results for two distinct orbits
  4. Results when one system is a rotation
  5. Results for a single orbit
  6. Two mixing systems proofs
  7. Proofs for the doubling map
  8. Proofs for Rotations
  9. Proofs for one orbit

Key Result

theorem 1.1

For a dynamical system $(X,T,\mu)$, we have

Theorems & Definitions (42)

  • theorem 1.1
  • theorem 2.1
  • definition 2.2
  • definition 2.3: see Sau00 and BarLiaRou19
  • theorem 2.4
  • remark 2.5
  • example 2.6
  • example 2.7
  • corollary 2.8
  • theorem 2.9
  • ...and 32 more