[Paper Review] Multifractal formalism derived from thermodynamics

Vaughn Climenhaga|arXiv (Cornell University)|Feb 3, 2010

Mathematical Dynamics and Fractals47 references3 citations

TL;DR

This paper establishes a rigorous connection between multifractal formalism and thermodynamic formalism by showing that multifractal spectra—such as those for pointwise dimension, local entropy, and Birkhoff averages—can be derived as Legendre transforms of thermodynamic functions $ T: \mathbb{R} \to \mathbb{R} $, under minimal smoothness assumptions, particularly that $ T $ is continuously differentiable. The key contribution is a general framework that reduces multifractal analysis to thermodynamic analysis, unifying and extending prior results in hyperbolic and non-uniformly hyperbolic systems.

ABSTRACT

We show that under quite general conditions, various multifractal spectra may be obtained as Legendre transforms of functions $T\colon \RR o \RR$ arising in the thermodynamic formalism. We impose minimal requirements on the maps we consider, and obtain partial results for any continuous map $f$ on a compact metric space. In order to obtain complete results, the primary hypothesis we require is that the functions $T$ be continuously differentiable. This makes rigorous the general paradigm of reducing questions regarding the multifractal formalism to questions regarding the thermodynamic formalism. These results hold for a broad class of measurable potentials, which includes (but is not limited to) continuous functions. We give applications that include most previously known results, as well as some new ones.

Motivation & Objective

• To establish a general theoretical framework linking multifractal formalism to thermodynamic formalism.
• To show that multifractal spectra (e.g., for pointwise dimension, local entropy, Birkhoff averages) arise as Legendre transforms of thermodynamic functions $ T $.
• To minimize assumptions on the system, requiring only continuity of the map and continuous differentiability of $ T $, to achieve broad applicability.
• To unify and generalize existing results on multifractal spectra in uniformly and non-uniformly hyperbolic systems.
• To provide a rigorous foundation for the paradigm that multifractal problems reduce to thermodynamic problems.

Proposed method

• Derive multifractal spectra $ \mathcal{D}(\alpha) $, $ \mathcal{E}(\alpha) $, and $ \mathcal{B}(\alpha) $ as Legendre transforms of thermodynamic functions $ T(q) $, using the duality between pressure functions and level sets of local quantities.
• Use the thermodynamic formalism to define pressure functions $ P^*(\varphi_{q,t}) = h(\nu) + q\int\varphi_1\,d\nu - t\lambda(\nu) $, where $ \nu $ is an invariant measure and $ \lambda(\nu) $ is the Lyapunov exponent.
• Apply the Implicit Function Theorem to construct continuously differentiable curves $ t = T_n(q) $ such that $ P^*(\varphi_{q,T_n(q)}) = 1/n $, ensuring convergence to the thermodynamic function $ T_{\mathcal{D}}(q) $.
• Use weak* convergence of empirical measures and the weak Gibbs property to show that invariant measures $ \nu_n $ associated with $ \varphi_{q_n,t_n} $ satisfy $ \int\varphi_1\,d\nu_n + \alpha\lambda(\nu_n) \to 0 $, implying $ \nu_n $-almost-everywhere convergence to level sets $ K_\alpha^{\mathcal{D}} $.
• Establish the key inequality $ \dim_H \nu_n \geq q_n\alpha + t_n $, leading to $ \mathcal{D}(\alpha) \geq q\alpha + T_{\mathcal{D}}(q) $, and conclude via Legendre duality that $ \mathcal{D}(\alpha) = T_{\mathcal{D}}^{L_3}(\alpha) $.
• Prove that $ T_{\mathcal{D}}(q) = \infty $ for $ q < 0 $ when $ \alpha = \infty $, using construction of invariant measures with zero entropy and negative $ \int\varphi_1\,d\nu $.

Experimental results

Research questions

• RQ1 Can multifractal spectra for pointwise dimension, local entropy, and Birkhoff averages be systematically derived from thermodynamic pressure functions?
• RQ2 Under what minimal conditions on the system and potential does the Legendre transform relationship between thermodynamic functions and multifractal spectra hold?
• RQ3 How does the continuous differentiability of the thermodynamic function $ T(q) $ ensure the validity and regularity of the multifractal spectrum?
• RQ4 Can the multifractal formalism be extended to non-uniformly hyperbolic systems using this thermodynamic framework?
• RQ5 What is the precise role of the pressure function $ P^*(\varphi_{q,t}) $ in characterizing the Hausdorff dimension of level sets of local quantities?

Key findings

• The multifractal spectrum $ \mathcal{D}(\alpha) $ for pointwise dimension is equal to the Legendre transform $ T_{\mathcal{D}}^{L_3}(\alpha) $ of the thermodynamic function $ T_{\mathcal{D}}(q) $, provided $ T_{\mathcal{D}} $ is continuously differentiable.
• For $ \alpha = \infty $, the spectrum satisfies $ T_{\mathcal{D}}(q) = \infty $ for all $ q < 0 $, which is consistent with the Legendre transform framework.
• The inequality $ \mathcal{D}(\alpha) \geq q\alpha + T_{\mathcal{D}}(q) $ holds for all $ q $, and equality is achieved in the limit via the construction of invariant measures $ \nu_n $ with $ \dim_H \nu_n \geq q_n\alpha + t_n $.
• The construction of measures $ \nu_n $ via pressure maximization ensures that $ \nu_n(K_\alpha^{\mathcal{D}}) = 1 $, and their dimension bounds yield the lower bound on $ \mathcal{D}(\alpha) $.
• The Implicit Function Theorem is applied to curves $ t = T_n(q) $ such that $ P^*(\varphi_{q,T_n(q)}) = 1/n $, and the derivative condition $ T_n'(q_n) = -\alpha $ ensures the existence of a measure $ \nu_n $ satisfying the key identity $ \int\varphi_1\,d\nu_n + \alpha\lambda(\nu_n) \to 0 $.
• The proof establishes that $ \mathcal{D}(\alpha) \geq T_{\mathcal{D}}^{L_3}(\alpha) $, and since the reverse inequality is standard, equality holds: $ \mathcal{D}(\alpha) = T_{\mathcal{D}}^{L_3}(\alpha) $, completing the Legendre duality.