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Conformal dynamics of fractal growth patterns without randomness
Davidovitch1, Feigenbaum, Hentschel
1Department of Chemical Physics, The Weizmann Institute of Science, Rehovot 76100, Israel.
Summary
This study introduces deterministic fractal growth models without randomness, using iterated conformal maps. These models enable a new scaling theory for understanding complex fractal patterns and their dimensions.
Area of Science:
- Complex Systems
- Mathematical Physics
- Fractal Geometry
Background:
- Fractal growth models often combine complex geometry with randomness, hindering analysis.
- Existing models like diffusion limited aggregation present significant challenges in theoretical elucidation.
Purpose of the Study:
- To introduce a novel class of fractal growth models that eliminate randomness.
- To analyze fractal patterns generated by deterministic dynamics of iterated conformal maps.
- To develop a scaling theory for these deterministic fractal growth patterns.
Main Methods:
- Defining models via deterministic itineraries of iterated conformal maps.
- Generating the conformal map function Phi((n))(omega) for n-particle aggregates.
- Focusing on quasiperiodic itineraries and their rational approximants.
- Applying analytic power to develop a scaling theory.
Main Results:
- Successfully generated complex fractal geometries using deterministic rules.
- Demonstrated that interface complexity arises from deterministic conformal map dynamics.
- Identified the exponent governing the fractal dimension through a developed scaling theory.
Conclusions:
- Deterministic models offer a tractable approach to studying complex fractal growth.
- Iterated conformal maps provide a powerful framework for generating and analyzing fractal structures.
- The developed scaling theory facilitates the prediction of fractal dimensions in these deterministic systems.
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