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<p>The property of exponential dichotomy can be seen as a generalization of the hyperbolicity condition for non autonomous linear finite dimensional systems of ordinary differential equations. In 1978 W.A. Coppel proved that the exponential dichotomy on the half line is equivalent to the property of noncritical uniformity provided that a condition of bounded growth is verified. In 2006 K.J. Palmer extended this result by proving that—also assuming the bounded growth condition—the properties of exponential dichotomy on the half line, noncritical uniformity and the exponential expansiveness are equivalent. The main contribution of this article is to generalize these results for the property of uniform <i>h</i>-dichotomy. This has been carried out due to a recent idea: any <i>h</i>-dichotomy can be associated to a totally ordered topological group, which becomes the additive group <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="208_2025_3281_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="44" /> </InlineMediaObject> <EquationSource Format="TEX">\((\mathbb {R},+)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="double-struck">R</mi> <mo>,</mo> <mo>+</mo> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> in case of the exponential dichotomy. The properties of this new group make possible such generalization.</p>

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Noncritical uniformity and expansiveness for uniform h-dichotomies

  • 期刊论文

  • 发表日期：

    2025年9月3日

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  • Heli Elorreaga,
  • Juan Francisco Peña,
  • Gonzalo Robledo

摘要

The property of exponential dichotomy can be seen as a generalization of the hyperbolicity condition for non autonomous linear finite dimensional systems of ordinary differential equations. In 1978 W.A. Coppel proved that the exponential dichotomy on the half line is equivalent to the property of noncritical uniformity provided that a condition of bounded growth is verified. In 2006 K.J. Palmer extended this result by proving that—also assuming the bounded growth condition—the properties of exponential dichotomy on the half line, noncritical uniformity and the exponential expansiveness are equivalent. The main contribution of this article is to generalize these results for the property of uniform h-dichotomy. This has been carried out due to a recent idea: any h-dichotomy can be associated to a totally ordered topological group, which becomes the additive group \((\mathbb {R},+)\) ( R , + ) in case of the exponential dichotomy. The properties of this new group make possible such generalization.