Bowen's Formula for a Dynamical Solenoid

Area of Science:

• Dynamical Systems
• Ergodic Theory
• Fractal Geometry

Background:

• Rufus Bowen established a formula linking ergodic theory and dimension theory in dynamical systems.
• The original Bowen's formula relates Hausdorff dimension of conformal repellers to pressure function zeros for a single conformal map.

Purpose of the Study:

• To extend Bowen's formula to a sequence of conformal maps.
• To investigate the applicability of Bowen's formula in the context of dynamical solenoids.

Main Methods:

• Definition of a dynamical solenoid using backward compositions of continuous surjections on a compact metric space.
• Development of a self-contained proof for the validity of Bowen's formula under mild assumptions.

Main Results:

• Demonstration that Bowen's formula holds for dynamical conformal solenoids.
• Establishment of the corollary that Bowen's formula is valid for a single conformal surjection on a compact space.

Conclusions:

• The findings generalize a fundamental result in dimension theory for dynamical systems.
• The research validates and extends the applicability of Bowen's formula to more complex systems like conformal solenoids.