[论文解读] On natural invariant measures on generalised iterated function systems

Antti Käenmäki|arXiv (Cornell University)|Jan 30, 2017

Mathematical Dynamics and Fractals参考文献 13被引用 65

一句话总结

该论文通过 cylinder functions 证明广义 IFS 的 t-平衡不变测度的存在，证明遍历性，并将平衡维度与极限集的 Hausdorff 维度联系起来，应用于自相似、自共形，以及几乎所有自仿射情形。

ABSTRACT

We consider the limit set of generalised iterated function systems. Under the assumption of a natural potential, the so called cylinder function, we prove the existence of the invariant probability measure satisfying the equilibrium state. We motivate this approach by showing that for typical self-affine sets there exists an ergodic invariant measure having the same Hausdorff dimension as the set itself.

研究动机与目标

• Motivate and generalize thermodynamical formalism to generalised iterated function systems (IFS).
• Define cylinder functions and a topological pressure to study invariant measures and equilibrium states.
• Establish existence and ergodicity of t-equilibrium measures and relate equilibrium dimension to Hausdorff dimension.
• Explore projections to limit sets via geometric projection and study separation conditions and dimension bounds.
• Specialize the framework to similitude, conformal, and affine IFS and discuss implications for Hausdorff dimensions.

提出的方法

• Introduce cylinder function psi_i^t on the symbol space I^∞ and define energy E_mu(t) and entropy h_mu.
• Define topological pressure P(t) and prove existence of t-equilibrium measure via generalized subadditivity.
• Prove ergodicity of the equilibrium measure using affine mapping properties and Choquet’s theorem.
• Develop a projection framework pi: I^∞ → X to obtain limit sets E and study their Hausdorff dimension via bi-Lipschitz and separation conditions.
• Apply framework to geometrically stable IFS (OSC/SSC) and show equality of equilibrium and Hausdorff dimensions in key cases.

实验结果

研究问题

• RQ1 Can we guarantee the existence of an invariant probability measure that attains the equilibrium state in a generalised IFS using cylinder functions?
• RQ2 Is the resulting equilibrium measure ergodic and does it have full equilibrium dimension (dim_ψ) matching zero of the pressure P(t)?
• RQ3 How do geometric projections of the symbol space to limit sets influence the Hausdorff dimension, and under what separation conditions can we bound it?
• RQ4 Do similitude and conformal IFS yield equilibrium measures whose dimension matches the limit set’s Hausdorff dimension, and what about almost all affine IFS?
• RQ5 What roles do geometrically stable IFS conditions (bounded overlap, bi-Lipschitz bounds) play in relating equilibrium and geometric dimensions?

主要发现

• Existence of a t-equilibrium measure for cylinder functions under generalized subadditivity.
• The t-equilibrium measure is ergodic and realizes full equilibrium dimension (dim_ψ) with P(t)=0.
• For geometrically stable IFS, one obtains natural upper and lower dimension bounds for the limit set from bi-Lipschitz constants and separation conditions.
• In the cases of similitude and conformal IFS, the equilibrium measure's Hausdorff dimension equals the limit set's Hausdorff dimension.
• For almost all affine IFS, the equilibrium dimension matches the Hausdorff dimension of the limit set, aligning with known results for self-affine sets.
• The framework recovers and extends known results for self-similar, self-conformal, and generic self-affine systems.