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<p>The aim of this paper is to reveal the relationship between the topological entropy of a non-autonomous iterated function system (or NAIFS for short) on countably infinite alphabets and that of its induced transformations. More precisely, we prove that the topological entropy of an NAIFS (<i>X</i>, Φ) vanishes if and only if the topological entropy of its induced system <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(({\cal{M}}(X),\Phi)\)</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mo stretchy="false">(</mo> <mrow> <mrow> <mi mathvariant="script">M</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mi mathvariant="normal">Φ</mi> <mo stretchy="false">)</mo> </math></EquationSource> </InlineEquation> is zero; if the topological entropy of (<i>X</i>, Φ) is positive, then that of its induced systems <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(({\cal{M}}(X),\Phi),({\cal{K}}(X),\Phi)\)</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mo stretchy="false">(</mo> <mrow> <mrow> <mi mathvariant="script">M</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mi mathvariant="normal">Φ</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mo stretchy="false">(</mo> <mrow> <mrow> <mi mathvariant="script">K</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mi mathvariant="normal">Φ</mi> <mo stretchy="false">)</mo> </math></EquationSource> </InlineEquation> are infinite.</p>

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On the Topological Entropy of Induced Transformations for Non-autonomous Iterated Function Systems

  • 期刊论文

  • 发表日期：

    2026年7月6日

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  • Lei Liu,
  • Cao Zhao

摘要

The aim of this paper is to reveal the relationship between the topological entropy of a non-autonomous iterated function system (or NAIFS for short) on countably infinite alphabets and that of its induced transformations. More precisely, we prove that the topological entropy of an NAIFS (X, Φ) vanishes if and only if the topological entropy of its induced system \(({\cal{M}}(X),\Phi)\) ( M ( X ) , Φ ) is zero; if the topological entropy of (X, Φ) is positive, then that of its induced systems \(({\cal{M}}(X),\Phi),({\cal{K}}(X),\Phi)\) ( M ( X ) , Φ ) , ( K ( X ) , Φ ) are infinite.