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<p>A dynamical system <InlineEquation ID="IEq1"> <EquationSource Format="MATHML"><math> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mi>T</mi> <mo stretchy="false">)</mo> </math></EquationSource> <EquationSource Format="TEX">$(X,T)$</EquationSource> </InlineEquation> is <i>shift embeddable</i> if <InlineEquation ID="IEq2"> <EquationSource Format="MATHML"><math> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mi>T</mi> <mo stretchy="false">)</mo> </math></EquationSource> <EquationSource Format="TEX">$(X,T)$</EquationSource> </InlineEquation> embeds continuously and equivariantly in the shift over <InlineEquation ID="IEq3"> <EquationSource Format="MATHML"><math> <msup> <mrow> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">]</mo> </mrow> <mi>d</mi> </msup> </math></EquationSource> <EquationSource Format="TEX">$[0,1]^{d}$</EquationSource> </InlineEquation> for some finite <InlineEquation ID="IEq4"> <EquationSource Format="MATHML"><math> <mi>d</mi> </math></EquationSource> <EquationSource Format="TEX">$d$</EquationSource> </InlineEquation>. Refuting a major conjecture in the field, in a recent result of Dranishnikov and Levin it was shown that Gromov’s mean dimension and Lebesgue covering dimension of finite orbits are not the only obstructions for shift embeddability. We present a new notion of dimension for dynamical systems over any countable group. We show that this new notion of dimension accounts for all known obstructions for shift embeddability.</p>

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A New Notion of Dimension for Dynamical Systems and Shift Embeddability

  • 期刊论文

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  • 发表日期：

    2026年6月17日

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  • Tom Meyerovitch

摘要

A dynamical system ( X , T ) $(X,T)$ is shift embeddable if ( X , T ) $(X,T)$ embeds continuously and equivariantly in the shift over [ 0 , 1 ] d $[0,1]^{d}$ for some finite d $d$ . Refuting a major conjecture in the field, in a recent result of Dranishnikov and Levin it was shown that Gromov’s mean dimension and Lebesgue covering dimension of finite orbits are not the only obstructions for shift embeddability. We present a new notion of dimension for dynamical systems over any countable group. We show that this new notion of dimension accounts for all known obstructions for shift embeddability.