We study the shortest distance, denoted by <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\widehat{m}_{n}(x,y)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mover accent="true"> <mi>m</mi> <mo stretchy="false">^</mo> </mover> <mi>n</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, between distinct n-step orbits of different dynamical systems. The main results concern the asymptotic behavior of <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\widehat{m}_{n}(x,y)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mover accent="true"> <mi>m</mi> <mo stretchy="false">^</mo> </mover> <mi>n</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> as n increases. We prove that the asymptotic behavior of <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\widehat{m}_{n}(x,y)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mover accent="true"> <mi>m</mi> <mo stretchy="false">^</mo> </mover> <mi>n</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> as n increases is characterized by the correlation dimension of the invariant measures of the systems, provided both systems exhibit exponential mixing in a suitable functional analytic framework. Applications of the results are provided for rotations and expanding maps.

The shortest distance between orbits of two different dynamical systems

期刊论文
发表日期：

2026年2月1日

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Saisai Shi,
Jia Liu,
Yuan Zhang

摘要

We study the shortest distance, denoted by $\widehat{m}_{n}(x,y)$ ${\hat{m}}_{n} (x, y)$ , between distinct n-step orbits of different dynamical systems. The main results concern the asymptotic behavior of $\widehat{m}_{n}(x,y)$ ${\hat{m}}_{n} (x, y)$ as n increases. We prove that the asymptotic behavior of $\widehat{m}_{n}(x,y)$ ${\hat{m}}_{n} (x, y)$ as n increases is characterized by the correlation dimension of the invariant measures of the systems, provided both systems exhibit exponential mixing in a suitable functional analytic framework. Applications of the results are provided for rotations and expanding maps.