Let (X, f) be a dynamical system possessing the specification property, and let <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\varphi $</EquationSource> <EquationSource Format="MATHML"><math> <mi>φ</mi> </math></EquationSource> </InlineEquation> be a continuous function. In this paper, we establish several conditional variational principles for the upper and lower Bowen/packing metric mean dimensions with potential, associated with the multifractal level set <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$K_\alpha := \{x \in X: \lim _{n \rightarrow \infty } \frac{1}{n} \sum _{i=0}^{n-1} \varphi (f^i x) = \alpha \}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>K</mi> <mi>α</mi> </msub> <mo>:</mo> <mo>=</mo> <mrow> <mo stretchy="false">{</mo> <mi>x</mi> <mo>∈</mo> <mi>X</mi> <mo>:</mo> <msub> <mo movablelimits="true">lim</mo> <mrow> <mi>n</mi> <mo stretchy="false">→</mo> <mi>∞</mi> </mrow> </msub> <mfrac> <mn>1</mn> <mi>n</mi> </mfrac> <msubsup> <mo>∑</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow> <mi>n</mi> <mo>-</mo> <mn>1</mn> </mrow> </msubsup> <mi>φ</mi> <mrow> <mo stretchy="false">(</mo> <msup> <mi>f</mi> <mi>i</mi> </msup> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mi>α</mi> <mo stretchy="false">}</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>.

Multifractal Level Sets and Metric Mean Dimension with Potential

期刊论文
发表日期：

2026年1月10日

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Tianlong Zhang,
Ercai Chen,
Xiaoyao Zhou

摘要

Let (X, f) be a dynamical system possessing the specification property, and let $\varphi $ $φ$ be a continuous function. In this paper, we establish several conditional variational principles for the upper and lower Bowen/packing metric mean dimensions with potential, associated with the multifractal level set $K_\alpha := \{x \in X: \lim _{n \rightarrow \infty } \frac{1}{n} \sum _{i=0}^{n-1} \varphi (f^i x) = \alpha \}$ $K_{α} : = {x \in X : {lim}_{n \to \infty} \frac{1}{n} \sum_{i = 0}^{n - 1} φ (f^{i} x) = α}$ .