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Entropy computing via integration over fractal measures.

Wojciech Słomczynski1, Jarosław Kwapien, Karol Zyczkowski

  • 1Instytut Matematyki, Uniwersytet Jagiellonski, ul. Reymonta 4, 30-059, Krakow, Poland.

Chaos (Woodbury, N.Y.)
|June 5, 2003
PubMed
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We explore invariant measures for iterated function systems (IFSs) with place-dependent probabilities, calculating Renyi entropies and multifractal spectra. This research introduces a novel method for computing entropy in dynamical systems by linking them to IFSs.

Area of Science:

  • Fractal Geometry
  • Dynamical Systems Theory
  • Information Theory

Background:

  • Iterated Function Systems (IFSs) are fundamental in fractal geometry.
  • Invariant measures are crucial for understanding the statistical properties of dynamical systems.
  • Place-dependent probabilities introduce complexity in traditional IFS models.

Purpose of the Study:

  • To investigate the properties of invariant measures for IFSs with place-dependent probabilities.
  • To compute key fractal and entropic characteristics: Renyi entropies, generalized dimensions, and multifractal spectra.
  • To establish a connection between the generalized entropies of dynamical systems and their associated IFSs.

Main Methods:

  • Analysis of invariant measures for place-dependent probability IFSs.

Related Experiment Videos

  • Computation of Renyi entropies, generalized dimensions, and multifractal spectra.
  • Numerical integration techniques applied to fractal measures.

    • Main Results:

    • Characterization of invariant measures for a generalized class of IFSs.
    • Demonstration that generalized entropies of certain dynamical systems can be made equal to those of associated IFSs.
    • Development of a new computational approach for entropy in dynamical systems.

    Conclusions:

    • The study provides a novel framework for analyzing complex fractal systems.
    • A new method for computing entropy in classical and quantum dynamical systems is presented.
    • The findings bridge concepts from fractal geometry and dynamical systems theory.