Multifractal Analysis of F-exponents for Finitely Irreducible Conformal Graph Directed Markov Systems (2404.09348v3)

Abstract: Let $\Phi = {\phi_e}_{e\in E}$ be a finitely irreducible conformal graph directed Markov system (CGDMS) with symbolic representation $E_A^{\infty}$ and limit set $J$. Under a mild condition on the system, we give a multifractal analysis of level sets of Birkhoff averages with respect to Hausdorff dimension for a large family of functions. We then apply these results to a few examples in the case of both $E$ finite and $E$ countably infinite.

• The paper introduces a novel approach to calculating the multifractal spectrum function for F-exponents using Hausdorff dimensions and Birkhoff averages.
• It demonstrates the existence and uniqueness of a real analytic solution for the spectrum under the Strong Open Set Condition and cofinite regularity criteria.
• The study reveals that the derivative analysis of the pressure function exposes non-convex spectral shapes, particularly in systems with infinite alphabets.

Multifractal Analysis of F-exponents for Finitely Irreducible Conformal Graph Directed Markov Systems

The paper under review presents a comprehensive multifractal analysis of F-exponents within the context of finitely irreducible conformal graph directed Markov systems (CGDMS). This research addresses the Hausdorff dimension of level sets of Birkhoff averages for a diverse class of functions. The exploration of these systems extends to cases with both finite and countably infinite alphabets, underpinning broader implications in the paper of dynamical systems and fractal geometry.

Theoretical Framework and Main Contributions

The paper begins by defining a CGDMS and sets the stage for a detailed examination of multifractality in the spectrum of F-exponents. By employing Birkhoff's Ergodic Theorem, the research discerns the characteristics of dynamical systems through their atypical sets, particularly under variations in Hausdorff dimension. Key to this exploration is the formulation of the pressure function associated with CGDMS and the utilization of conformal structures to facilitate invariant measures.

Key Results:

1. Multifractal Spectrum Function: The paper introduces a unique approach to calculating the multifractal spectrum function $t(\xi)$ for F-exponents. The authors leverage Hausdorff dimensions to outline this function's dependence on Birkhoff averages in both the symbolic representation and the limit set.
2. Existence and Uniqueness: It is demonstrated that the system of equations describing the multifractal spectrum admits a unique real analytic solution within the multifractal region $D_0$. This solution holds for CGDMS that satisfy the Strong Open Set Condition (SOSC) and cofinite regularity, extending previous studies' applicability.
3. Convergence of Gibbs States: A significant methodological contribution of this paper is the demonstration of Gibbs states' tightness and their convergence, allowing for a zero-temperature type limit analysis. This has implications for understanding equilibrium states within these systems and paves the way for further analytical work in multifractal analysis.
4. Derivative Analysis and Spectral Shape: A detailed analysis of the derivative properties of the pressure function highlights the spectrum's shape, revealing non-convexity for infinite alphabet cases. This insight is crucial for understanding the nuanced geometric structure of the spectrum, especially concerning the Lyapunov exponent.

Implications and Future Directions

The implications of these findings span both theoretical and practical domains. Theoretically, the results enrich the understanding of multifractal structures in dynamical systems, particularly those with complex symbolic dynamics. This work advances the field by providing a robust analytical framework to explore these systems' intricate properties, setting the stage for subsequent studies of more complex or higher-dimensional systems.

On a practical note, the insights gleaned from such multifractal analyses have potential applications in fields ranging from statistical mechanics to data science, where understanding the distribution and scaling properties of complex systems can be invaluable.

Future research could extend the current analysis to consider non-conformal settings or systems with additional constraints or symmetries. Furthermore, the methodology might be adapted to explore other spectral types, enhancing the breadth and scope of multifractal analysis.

In summary, this paper provides a vital contribution to fractal geometry and dynamical systems through its rigorous analysis of F-exponents within CGDMS, offering new tools and perspectives for researchers engaged in the multifractal characterization of complex systems.