Multifractal structure of a riddled basin.

Area of Science:

• Dynamical Systems and Chaos Theory
• Fractal Geometry
• Nonlinear Dynamics

Background:

• Riddled basins are complex structures in two-dimensional piecewise linear maps.
• Characterizing the boundaries of these basins is crucial for understanding their dynamics.
• Existing methods struggle to fully describe the intricate nature of riddled basins.

Purpose of the Study:

• To introduce a multifractal spectrum, f(gamma), for characterizing the skeletons of riddled basins.
• To derive the uncertainty exponent using a variational principle based on f(gamma).
• To propose a concept of a boundary for riddled basins and explore its relation to stable sets.

Main Methods:

• Development of a multifractal spectrum, f(gamma), tailored for riddled basins.
• Application of a variational principle to obtain the uncertainty exponent from f(gamma).
• Analysis of the relationship between f(gamma) and stable sets of ergodic measures.

Main Results:

• The multifractal spectrum f(gamma) effectively characterizes the skeletons of riddled basins.
• A variational principle yields the uncertainty exponent, enabling a boundary definition.
• A conjecture is proposed linking f(gamma) to stable sets of coexisting ergodic measures.

Conclusions:

• The multifractal spectrum provides a novel tool for analyzing complex dynamical systems.
• The concept of a boundary for riddled basins is established through the uncertainty exponent.
• Further research is suggested on the interplay between multifractality and ergodic measures in chaotic attractors.