<p>In this paper we calculate the metric and folding entropies for a family of non-invertible symbolic dynamical systems <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$(\Sigma _{m_-,m_+}, \sigma _\phi )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <msub> <mi mathvariant="normal">Σ</mi> <mrow> <msub> <mi>m</mi> <mo>-</mo> </msub> <mo>,</mo> <msub> <mi>m</mi> <mo>+</mo> </msub> </mrow> </msub> <mo>,</mo> <msub> <mi>σ</mi> <mi>ϕ</mi> </msub> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> which generalizes the standard bilateral Bernoulli shifts. The space <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\Sigma _{m_-,m_+}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="normal">Σ</mi> <mrow> <msub> <mi>m</mi> <mo>-</mo> </msub> <mo>,</mo> <msub> <mi>m</mi> <mo>+</mo> </msub> </mrow> </msub> </math></EquationSource> </InlineEquation> consists of symbolic sequences over two distinct finite alphabets, with dynamics governed by a shift map <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\sigma _\phi $</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>σ</mi> <mi>ϕ</mi> </msub> </math></EquationSource> </InlineEquation> incorporating a non-invertible function <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\phi $</EquationSource> <EquationSource Format="MATHML"><math> <mi>ϕ</mi> </math></EquationSource> </InlineEquation> that maps one of the alphabets to the other one. These systems are, for instance, particularly useful for encoding the many-to-one baker’s transformation endomorphisms, and they can also be seen as a skew product with a unilateral Bernoulli shift on the base.</p>

Folding and Metric Entropies for Extended Shifts

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发表日期：

2026年1月12日

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Neemias Martins,
Pedro G. Mattos,
Régis Varão

摘要

In this paper we calculate the metric and folding entropies for a family of non-invertible symbolic dynamical systems $(\Sigma _{m_-,m_+}, \sigma _\phi )$ $(Σ_{m_{-}, m_{+}}, σ_{ϕ})$ which generalizes the standard bilateral Bernoulli shifts. The space $\Sigma _{m_-,m_+}$ $Σ_{m_{-}, m_{+}}$ consists of symbolic sequences over two distinct finite alphabets, with dynamics governed by a shift map $\sigma _\phi $ $σ_{ϕ}$ incorporating a non-invertible function $\phi $ $ϕ$ that maps one of the alphabets to the other one. These systems are, for instance, particularly useful for encoding the many-to-one baker’s transformation endomorphisms, and they can also be seen as a skew product with a unilateral Bernoulli shift on the base.