<p>We aim to investigate the dimension theory of <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\alpha $</EquationSource> <EquationSource Format="MATHML"><math> <mi>α</mi> </math></EquationSource> </InlineEquation>-pressure-like quantities. By means of the Carath<InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\acute{\textrm{e}}$</EquationSource> <EquationSource Format="MATHML"><math> <mover accent="true"> <mtext>e</mtext> <mo>´</mo> </mover> </math></EquationSource> </InlineEquation>odory-Pesin structure, we define <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\alpha $</EquationSource> <EquationSource Format="MATHML"><math> <mi>α</mi> </math></EquationSource> </InlineEquation>-BS dimension and <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\alpha $</EquationSource> <EquationSource Format="MATHML"><math> <mi>α</mi> </math></EquationSource> </InlineEquation>-Pesin topological pressure on subsets using <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$\alpha $</EquationSource> <EquationSource Format="MATHML"><math> <mi>α</mi> </math></EquationSource> </InlineEquation>-Bowen metric <Equation ID="Equ5"> <EquationSource Format="TEX">$\begin{aligned} d_{n}^{\alpha }(x,y)=\max _{0\le i\le n-1}e^{\alpha i}d(f^{i}x,f^{i}y), \end{aligned}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mrow> <msubsup> <mi>d</mi> <mrow> <mi>n</mi> </mrow> <mi>α</mi> </msubsup> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <munder> <mo movablelimits="true">max</mo> <mrow> <mn>0</mn> <mo>≤</mo> <mi>i</mi> <mo>≤</mo> <mi>n</mi> <mo>-</mo> <mn>1</mn> </mrow> </munder> <msup> <mi>e</mi> <mrow> <mi>α</mi> <mi>i</mi> </mrow> </msup> <mi>d</mi> <mrow> <mo stretchy="false">(</mo> <msup> <mi>f</mi> <mi>i</mi> </msup> <mi>x</mi> <mo>,</mo> <msup> <mi>f</mi> <mi>i</mi> </msup> <mi>y</mi> <mo stretchy="false">)</mo> </mrow> <mo>,</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>where <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$\alpha \ge 0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>α</mi> <mo>≥</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>. Specifically, we show that <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$\alpha $</EquationSource> <EquationSource Format="MATHML"><math> <mi>α</mi> </math></EquationSource> </InlineEquation>-BS dimension and <InlineEquation ID="IEq11"> <EquationSource Format="TEX">$\alpha $</EquationSource> <EquationSource Format="MATHML"><math> <mi>α</mi> </math></EquationSource> </InlineEquation>-Pesin topological pressure are related by a Bowen’s equation. Inspired by the classical Brin–Katok entropy, we introduce the notion of <InlineEquation ID="IEq12"> <EquationSource Format="TEX">$\alpha $</EquationSource> <EquationSource Format="MATHML"><math> <mi>α</mi> </math></EquationSource> </InlineEquation>-local Brin–Katok entropy, and establish a variational principle for <InlineEquation ID="IEq13"> <EquationSource Format="TEX">$\alpha $</EquationSource> <EquationSource Format="MATHML"><math> <mi>α</mi> </math></EquationSource> </InlineEquation>-BS dimension on compact subsets in terms of <InlineEquation ID="IEq14"> <EquationSource Format="TEX">$\alpha $</EquationSource> <EquationSource Format="MATHML"><math> <mi>α</mi> </math></EquationSource> </InlineEquation>-local Brin–Katok entropy. Besides, for subshifts of finite type, we prove that <InlineEquation ID="IEq15"> <EquationSource Format="TEX">$\alpha $</EquationSource> <EquationSource Format="MATHML"><math> <mi>α</mi> </math></EquationSource> </InlineEquation>-Bowen topological entropy is closely related to spectral radius and Hausdorff dimension.</p>

$\alpha $-BS dimension on subsets

期刊论文
发表日期：

2026年1月4日

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Zhumin Ding,
Rui Yang,
Xiaoyao Zhou

摘要

We aim to investigate the dimension theory of $\alpha $ $α$ -pressure-like quantities. By means of the Carath $\acute{\textrm{e}}$ $\overset{´}{e}$ odory-Pesin structure, we define $\alpha $ $α$ -BS dimension and $\alpha $ $α$ -Pesin topological pressure on subsets using $\alpha $ $α$ -Bowen metric $\begin{aligned} d_{n}^{\alpha }(x,y)=\max _{0\le i\le n-1}e^{\alpha i}d(f^{i}x,f^{i}y), \end{aligned}$ $\begin{matrix} d_{n}^{α} (x, y) = max_{0 \leq i \leq n - 1} e^{α i} d (f^{i} x, f^{i} y), \end{matrix}$ where $\alpha \ge 0$ $α \geq 0$ . Specifically, we show that $\alpha $ $α$ -BS dimension and $\alpha $ $α$ -Pesin topological pressure are related by a Bowen’s equation. Inspired by the classical Brin–Katok entropy, we introduce the notion of $\alpha $ $α$ -local Brin–Katok entropy, and establish a variational principle for $\alpha $ $α$ -BS dimension on compact subsets in terms of $\alpha $ $α$ -local Brin–Katok entropy. Besides, for subshifts of finite type, we prove that $\alpha $ $α$ -Bowen topological entropy is closely related to spectral radius and Hausdorff dimension.

\(\alpha \)-BS dimension on subsets

摘要