A bivariate interaction scheme is introduced based on single-humped functions f(x) and g(y), which generate the length scales and their measures, respectively, according to a multifractal distribution. The largest eigenvalue of this procedure is connected with the Gibbs potential G(q), from which dimensions, entropies, and Lyapunov exponents can be extracted. The mechanisms leading to phase transitions in the Gibbs potential and in multifractal spectra are analyzed. The method provides us with a general scheme for classifying possible phase transitions in multifractals. As a novel phenomenon in the field of dynamical systems, we study phase transitions in spectra belonging to the natural measure of nonhyperbolic repellers of one-dimensional maps.
ASJC Scopus subject areas
- Atomic and Molecular Physics, and Optics