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<p>In this paper, we investigate the double ergodicity of strong horseshoe maps, defined as onto maps whose phase spaces act as attractors for their inverse iterations. We prove that such maps, when possessing the reverse bounded distortion property, are doubly ergodic with respect to the Lebesgue measure. Additionally, we establish the robustness of double ergodicity and weak mixing for a <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(C^1\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>C</mi> <mn>1</mn> </msup> </math></EquationSource> </InlineEquation>-perturbed doubling map on the circle, demonstrating that all maps in a <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(C^1\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>C</mi> <mn>1</mn> </msup> </math></EquationSource> </InlineEquation>-neighborhood, as given in Theorem <InternalRef RefID="FPar2">B</InternalRef>, share these properties.</p>

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Double ergodicity of strong horseshoe maps

  • 期刊论文

  • 发表日期：

    2026年3月6日

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  • Aliasghar Sarizadeh

摘要

In this paper, we investigate the double ergodicity of strong horseshoe maps, defined as onto maps whose phase spaces act as attractors for their inverse iterations. We prove that such maps, when possessing the reverse bounded distortion property, are doubly ergodic with respect to the Lebesgue measure. Additionally, we establish the robustness of double ergodicity and weak mixing for a \(C^1\) C 1 -perturbed doubling map on the circle, demonstrating that all maps in a \(C^1\) C 1 -neighborhood, as given in Theorem B, share these properties.