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Bowen's Formula for a Dynamical Solenoid.

Andrzej Biś1, Wojciech Kozłowski1, Agnieszka Marczuk1

  • 1Faculty of Mathematics and Computer Science, University of Lodz, Banacha 22, 90-238 Łódź, Poland.

Entropy (Basel, Switzerland)
|November 27, 2024
PubMed
Summary
This summary is machine-generated.

This study extends Bowen's formula, connecting ergodic theory and dimension theory, to sequences of conformal maps. The research proves the formula holds for dynamical conformal solenoids, a generalized dynamical system.

Keywords:
conformal mapstopological entropytopological pressure

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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.