<p>In this paper we give a fully combinatorial description of the zero entropy periodic patterns on trees. Unlike previously known characterizations of such patterns, our criterion is independent of any particular topological realization of the pattern and provides, thus, a practical and fast algorithm to test zero entropy. As an application, consider a <InlineEquation ID="IEq1"> <EquationSource Format="TEX">${\textbf {k}}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="bold">k</mi> </math></EquationSource> </InlineEquation>-star <InlineEquation ID="IEq2"> <EquationSource Format="TEX">${\textbf {T}}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="bold">T</mi> </math></EquationSource> </InlineEquation> (a tree with <InlineEquation ID="IEq3"> <EquationSource Format="TEX">${\textbf {k}}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="bold">k</mi> </math></EquationSource> </InlineEquation> edges attached at a unique branching point of valence <InlineEquation ID="IEq4"> <EquationSource Format="TEX">${\textbf {k}}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="bold">k</mi> </math></EquationSource> </InlineEquation>) and the set <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\mathcal {F}_{n,k}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="script">F</mi> <mrow> <mi>n</mi> <mo>,</mo> <mi>k</mi> </mrow> </msub> </math></EquationSource> </InlineEquation> of all continuous maps <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$f:{T} \longrightarrow {T}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>f</mi> <mo>:</mo> <mi>T</mi> <mo stretchy="false">⟶</mo> <mi>T</mi> </mrow> </math></EquationSource> </InlineEquation> having a periodic orbit of period <i>n</i> properly contained in <InlineEquation ID="IEq7"> <EquationSource Format="TEX">${\textbf {T}}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="bold">T</mi> </math></EquationSource> </InlineEquation> (each edge of <InlineEquation ID="IEq8"> <EquationSource Format="TEX">${\textbf {T}}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="bold">T</mi> </math></EquationSource> </InlineEquation> contains at least one point of the orbit). We find all pairs <InlineEquation ID="IEq9"> <EquationSource Format="TEX">${\textbf {(n,k)}}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="bold">n</mi> <mo>,</mo> <mi mathvariant="bold">k</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> such that <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$\mathcal {F}_{n,k}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="script">F</mi> <mrow> <mi>n</mi> <mo>,</mo> <mi>k</mi> </mrow> </msub> </math></EquationSource> </InlineEquation> contains maps of entropy zero, and we describe the patterns of such zero-entropy orbits.</p>

Zero Entropy Cycles on Trees: From Topology To Combinatorics And An Application To Star Maps

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2026年7月3日

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David Juher,
Francesc Mañosas,
David Rojas

摘要

In this paper we give a fully combinatorial description of the zero entropy periodic patterns on trees. Unlike previously known characterizations of such patterns, our criterion is independent of any particular topological realization of the pattern and provides, thus, a practical and fast algorithm to test zero entropy. As an application, consider a ${\textbf {k}}$ $k$ -star ${\textbf {T}}$ $T$ (a tree with ${\textbf {k}}$ $k$ edges attached at a unique branching point of valence ${\textbf {k}}$ $k$ ) and the set $\mathcal {F}_{n,k}$ $F_{n, k}$ of all continuous maps $f:{T} \longrightarrow {T}$ $f : T ⟶ T$ having a periodic orbit of period n properly contained in ${\textbf {T}}$ $T$ (each edge of ${\textbf {T}}$ $T$ contains at least one point of the orbit). We find all pairs ${\textbf {(n,k)}}$ $(n, k)$ such that $\mathcal {F}_{n,k}$ $F_{n, k}$ contains maps of entropy zero, and we describe the patterns of such zero-entropy orbits.