[Paper Review] Multifractal Analysis of The New Level Sets

Yiwei Dong, Xueting Tian|arXiv (Cornell University)|Oct 22, 2015

Mathematical Dynamics and Fractals14 references4 citations

TL;DR

This paper introduces a new multifractal analysis framework by partitioning the irregular set into level sets based on distinct liminf and limsup Birkhoff averages. Under the almost specification property, it establishes variational principles for topological entropy of these refined level sets, showing that entropy is determined by suprema of measure-theoretic entropies over invariant measures with prescribed averages, extending classical results to irregular and mixed regular-irregular sets.

ABSTRACT

By an appropriate definition, we divide the irregular set into level sets. Then we characterize the multifractal spectrum of these new pieces by calculating their entropies. We also compute the entropies of various intersections of the level sets of regular and irregular set which is rarely studied in the literature. Moreover, our conclusions also hold for the topological pressure. Finally, we consider the continuous case and use our results to give a description for the suspension flow.

Motivation & Objective

• To refine the multifractal decomposition of dynamical systems by introducing level sets based on distinct liminf and limsup Birkhoff averages.
• To characterize the topological entropy of these new level sets, especially for intersections of regular and irregular sets.
• To extend variational principles to the continuous case using suspension flows.
• To establish a connection between the entropy of level sets and invariant measures via suprema of measure-theoretic entropies.
• To generalize classical multifractal results to systems with the almost specification property, which strictly extend the specification property.

Proposed method

• Define new level sets $X_\varphi(c,d) = \{x \in X : \underline{\varphi}(x) = c, \overline{\varphi}(x) = d\}$ for $c \leq d$, partitioning the space based on distinct lower and upper Birkhoff averages.
• Use the almost specification property to ensure the existence of orbit gluing, enabling construction of measures with prescribed ergodic averages.
• Apply the variational principle for topological pressure, relating the entropy of level sets to suprema of $h_\mu(T) + \int \psi \, d\mu$ over invariant measures with fixed average values.
• Establish a correspondence between the suspension flow of a discrete system and the continuous flow, using conjugacy via scaling of return times.
• Prove that for any $c \leq d$, either $X_\varphi(c,d)$ is empty or its topological entropy equals $\min_{\xi = c,d} \sup \{ h_\mu(T) : \int \varphi \, d\mu = \xi \}$.
• Use the conjugacy between suspension flows with different roof functions to reduce the general case to the constant roof function case, simplifying entropy estimation.

Experimental results

Research questions

• RQ1 How can the irregular set be refined into level sets using distinct liminf and limsup Birkhoff averages, and what is the entropy of these sets?
• RQ2 What is the topological entropy of intersections between regular sets $R_{\varphi_1}(a)$ and irregular sets $I_{\varphi_2}$, and when is it non-empty?
• RQ3 Can a variational principle be established for the entropy of these new level sets under the almost specification property?
• RQ4 How does the multifractal spectrum of these refined level sets relate to invariant measures and their entropies?
• RQ5 Can the results be extended to continuous-time systems via suspension flows, and how does the roof function affect the entropy of level sets?

Key findings

• For any continuous functions $\varphi_1, \varphi_2$, the intersection $R_{\varphi_1}(a) \cap I_{\varphi_2}$ has full topological entropy or is empty.
• If $R_{\varphi_1}(a) \cap I_{\varphi_2} \neq \emptyset$, then its topological entropy equals $\sup \{ h_\mu(T) : \int \varphi_1 \, d\mu = a \}$, extending the classical variational principle to mixed regular-irregular sets.
• For any $c \leq d$, the topological entropy of $X_\varphi(c,d)$ is $\min_{\xi = c,d} \sup \{ h_\mu(T) : \int \varphi \, d\mu = \xi \}$, provided the set is non-empty.
• The entropy of the level set $X_{\varphi,\rho}(c,d)$ for weighted Birkhoff averages satisfies $P_{X_{\varphi,\rho}(c,d)}(\psi) = \min_{\xi = c,d} \sup \{ h_\mu(T) + \int \psi \, d\mu : \frac{\int \varphi \, d\mu}{\int \rho \, d\mu} = \xi \}$.
• For suspension flows, the topological entropy of level sets $X^\rho_\Phi(\Psi,c,d)$ is given by $\min_{\xi = c,d} \sup \{ h_{\mu_\rho}(T) : \int \Phi \, d\mu_\rho = \xi \}$, where $\mu_\rho$ is the induced measure on the suspended system.
• The results hold for topological pressure, and the entropy of level sets in the suspension flow is preserved under conjugacy, even when the roof function is non-constant.