<p>In this article, we investigate the <InlineEquation ID="IEq3"> <EquationSource Format="TEX">${\mathbb {Z}}^d$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">Z</mi> </mrow> <mi>d</mi> </msup> </math></EquationSource> </InlineEquation>-action <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\Phi$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Φ</mi> </math></EquationSource> </InlineEquation> on the Banach space such that each generator of <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\Phi$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Φ</mi> </math></EquationSource> </InlineEquation> consists of a linear part and a perturbed part. By adding certain conditions for the linear and perturbed parts of the generator, the notions Lipschitz hyperbolic <InlineEquation ID="IEq6"> <EquationSource Format="TEX">${\mathbb {Z}}^d$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">Z</mi> </mrow> <mi>d</mi> </msup> </math></EquationSource> </InlineEquation>-action and the strong partially hyperbolic <InlineEquation ID="IEq7"> <EquationSource Format="TEX">${\mathbb {Z}}^d$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">Z</mi> </mrow> <mi>d</mi> </msup> </math></EquationSource> </InlineEquation>-action are introduced. We show that <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$\Phi$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Φ</mi> </math></EquationSource> </InlineEquation> has the shadowing (quasi-stability) property when <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$\Phi$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Φ</mi> </math></EquationSource> </InlineEquation> is a Lipschitz hyperbolic (strong partially hyperbolic) <InlineEquation ID="IEq10"> <EquationSource Format="TEX">${\mathbb {Z}}^d$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">Z</mi> </mrow> <mi>d</mi> </msup> </math></EquationSource> </InlineEquation>-action.</p>

Quasi-stability and shadowing dynamics for ${\mathbb {Z}}^d$-actions on Banach spaces

期刊论文
发表日期：

2026年1月4日

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Peirong Li,
Zhiming Li,
Bilel Selmi

摘要

In this article, we investigate the ${\mathbb {Z}}^d$ $Z^{d}$ -action $\Phi$ $Φ$ on the Banach space such that each generator of $\Phi$ $Φ$ consists of a linear part and a perturbed part. By adding certain conditions for the linear and perturbed parts of the generator, the notions Lipschitz hyperbolic ${\mathbb {Z}}^d$ $Z^{d}$ -action and the strong partially hyperbolic ${\mathbb {Z}}^d$ $Z^{d}$ -action are introduced. We show that $\Phi$ $Φ$ has the shadowing (quasi-stability) property when $\Phi$ $Φ$ is a Lipschitz hyperbolic (strong partially hyperbolic) ${\mathbb {Z}}^d$ $Z^{d}$ -action.

Quasi-stability and shadowing dynamics for \({\mathbb {Z}}^d\)-actions on Banach spaces

摘要