[Paper Review] Multifractal analysis for multimodal maps

Mike Todd|ArXiv.org|Sep 5, 2008

Mathematical Dynamics and Fractals50 references16 citations

TL;DR

This paper develops a multifractal analysis for multimodal interval maps with critical points using inducing schemes based on the Hofbauer tower. By showing that critical orbits are sufficiently captured by these schemes, the authors establish the dimension spectrum and Lyapunov spectrum for equilibrium states of Hölder potentials, with optimal results under Collet-Eckmann conditions.

ABSTRACT

Given a multimodal interval map $f:I o I$ and a Hölder potential $ϕ:I o \mathbb{R}$, we study the dimension spectrum for equilibrium states of $ϕ$. The main tool here is inducing schemes, used to overcome the presence of critical points. The key issue is to show that enough points are `seen' by a class of inducing schemes. We also compute the Lyapunov spectrum. We obtain the strongest results when $f$ is a Collet-Eckmann map, but our analysis also holds for maps satisfying much weaker growth conditions.

Motivation & Objective

• To extend multifractal analysis to multimodal interval maps with critical points, where standard Markov coding fails due to non-uniform hyperbolicity.
• To address the challenge that standard inducing schemes (e.g., first return maps) may miss significant portions of the phase space when critical orbits are dense.
• To develop a framework using Hofbauer tower-based inducing schemes that ensure sufficient coverage of points with large Lyapunov exponents.
• To compute the dimension spectrum and Lyapunov spectrum for equilibrium states of Hölder potentials on such maps.
• To establish uniform bounds on the Radon-Nikodym derivative between equilibrium and conformal measures, enabling application of thermodynamic formalism on countable Markov shifts.

Proposed method

• Utilizes the Hofbauer tower (Markov extension) to construct inducing schemes that capture points with large pointwise Lyapunov exponents.
• Applies inducing schemes of type A and B, ensuring compatibility with the thermodynamic formalism for countable Markov shifts.
• Employs the theory of conformal measures and equilibrium states via the pressure function, with induced potentials satisfying $ P( ilde{ heta}) = 0 $.
• Uses Kac’s lemma and Hölder continuity to bound the partition function $ Z_0( ilde{ heta}) $, ensuring convergence of the pressure.
• Establishes uniform bounds on the density $ \frac{d\mu_\varphi}{dm_\varphi} $, both above and below, via comparison with induced measures.
• Applies results from Iommi (2009) on multifractal analysis for countable Markov shifts to derive the dimension spectrum $ \mathcal{DS}_\varphi(\alpha) $.

Experimental results

Research questions

• RQ1 Can the dimension spectrum of equilibrium states be computed for multimodal maps with critical points, where standard Markov coding fails?
• RQ2 How can inducing schemes be constructed to ensure that points with large Lyapunov exponents are captured, despite dense critical orbits?
• RQ3 Under what conditions does the induced potential satisfy $ P(\Phi) = 0 $, enabling application of thermodynamic formalism?
• RQ4 What is the relationship between the Radon-Nikodym derivative $ \frac{d\mu_\varphi}{dm_\varphi} $ and the induced measure $ \mu_\Phi $, and is it uniformly bounded?
• RQ5 How does the Lyapunov spectrum relate to the multifractal structure of the system under Hölder potentials?

Key findings

• The dimension spectrum $ \mathcal{DS}_\varphi(\alpha) $ is well-defined and computable for equilibrium states of Hölder potentials on multimodal maps.
• For Collet-Eckmann maps, the multifractal analysis yields the strongest results, with full regularity of the spectrum.
• The density $ \frac{d\mu_\varphi}{dm_\varphi} $ is uniformly bounded above and below, ensuring compatibility between the original and induced systems.
• The induced potential $ \Phi $ satisfies $ P(\Phi) = 0 $ for all inducing schemes in $ SCover^B(\varepsilon) $, enabling use of thermodynamic formalism.
• The set of points not seen by the inducing schemes is negligible in the sense of measure, provided the critical orbits satisfy mild growth conditions.
• The Lyapunov spectrum is computed via the same framework, with the key insight that points with zero Lyapunov exponent are not captured by the inducing method, suggesting optimality of the approach.