<p>Based on the <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\alpha $</EquationSource> <EquationSource Format="MATHML"><math> <mi>α</mi> </math></EquationSource> </InlineEquation>-metric <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$d_{n}^{\alpha }(x,y)=\max _{0\le i< n}e^{i\alpha }d(f^{i}x,f^{i}y),$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mi>d</mi> <mrow> <mi>n</mi> </mrow> <mi>α</mi> </msubsup> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <msub> <mo movablelimits="true">max</mo> <mrow> <mn>0</mn> <mo>≤</mo> <mi>i</mi> <mo><</mo> <mi>n</mi> </mrow> </msub> <msup> <mi>e</mi> <mrow> <mi>i</mi> <mi>α</mi> </mrow> </msup> <mi>d</mi> <mrow> <mo stretchy="false">(</mo> <msup> <mi>f</mi> <mi>i</mi> </msup> <mi>x</mi> <mo>,</mo> <msup> <mi>f</mi> <mi>i</mi> </msup> <mi>y</mi> <mo stretchy="false">)</mo> </mrow> <mo>,</mo> </mrow> </math></EquationSource> </InlineEquation> where <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\alpha \ge 0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>α</mi> <mo>≥</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>, we introduce <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\alpha $</EquationSource> <EquationSource Format="MATHML"><math> <mi>α</mi> </math></EquationSource> </InlineEquation>-estimation BS-Packing dimension and <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$\alpha $</EquationSource> <EquationSource Format="MATHML"><math> <mi>α</mi> </math></EquationSource> </InlineEquation>-estimation packing topological pressure on subsets by the Carathéodory structure. Motivated by the classical Brin-Katok entropy, we define several measure-theoretic quantities and derive a variational principle for <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$\alpha $</EquationSource> <EquationSource Format="MATHML"><math> <mi>α</mi> </math></EquationSource> </InlineEquation>-estimation BS-Packing dimension. We show that <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$\alpha $</EquationSource> <EquationSource Format="MATHML"><math> <mi>α</mi> </math></EquationSource> </InlineEquation>-estimation BS-Packing dimension and <InlineEquation ID="IEq11"> <EquationSource Format="TEX">$\alpha $</EquationSource> <EquationSource Format="MATHML"><math> <mi>α</mi> </math></EquationSource> </InlineEquation>-estimation packing topological pressure are connected via Bowen’s equation. Additionally, we explore connections between <InlineEquation ID="IEq12"> <EquationSource Format="TEX">$\alpha $</EquationSource> <EquationSource Format="MATHML"><math> <mi>α</mi> </math></EquationSource> </InlineEquation>-estimation packing topological pressure and other <InlineEquation ID="IEq13"> <EquationSource Format="TEX">$\alpha $</EquationSource> <EquationSource Format="MATHML"><math> <mi>α</mi> </math></EquationSource> </InlineEquation>-pressure-like quantities.</p>

$\alpha $-estimation BS-Packing Dimension and Bowen’s Equation

期刊论文
发表日期：

2026年3月28日

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Fei Gao,
Ercai Chen,
Yunxiang Xie

摘要

Based on the $\alpha $ $α$ -metric $d_{n}^{\alpha }(x,y)=\max _{0\le i< n}e^{i\alpha }d(f^{i}x,f^{i}y),$ $d_{n}^{α} (x, y) = {max}_{0 \leq i < n} e^{i α} d (f^{i} x, f^{i} y),$ where $\alpha \ge 0$ $α \geq 0$ , we introduce $\alpha $ $α$ -estimation BS-Packing dimension and $\alpha $ $α$ -estimation packing topological pressure on subsets by the Carathéodory structure. Motivated by the classical Brin-Katok entropy, we define several measure-theoretic quantities and derive a variational principle for $\alpha $ $α$ -estimation BS-Packing dimension. We show that $\alpha $ $α$ -estimation BS-Packing dimension and $\alpha $ $α$ -estimation packing topological pressure are connected via Bowen’s equation. Additionally, we explore connections between $\alpha $ $α$ -estimation packing topological pressure and other $\alpha $ $α$ -pressure-like quantities.

\(\alpha \)-estimation BS-Packing Dimension and Bowen’s Equation

摘要