Generalized dimensions, large deviations and the distribution of rare events

Any fractal set possesses a unique dimension, which characterises its self-similar behaviour under a change of scale. The attractors of many dynamical systems are more complex multifractals, which possess a spectrum of dimensions, or “generalized” dimensions. Interest in these quantities originated in the 1980s, primarily for the study of chaotic attractors and turbulence, and many techniques exist to calculate these dimensions numerically. In a new paper, LML External Fellow Davide Faranda and colleagues link generalized dimensions to the recurrence properties of a dynamical system, building on earlier work which showed that generalized dimensions can be derived from the moments of the so–called first return time, the length of time required for the dynamics to return close to a chosen initial point on an attractor.
The researchers also report a related result by considering hitting times, rather than return times. Hitting times reflect how long it takes to reach a small region of phase space starting from an arbitrary point. The authors demonstrate that generalized dimensions yield the rate function for the large deviations of the first hitting time of a ball of given radius, quantifying the rate at which the probability of observing “non-typical” values diminishes when the radius goes to zero.
Faranda and colleagues suggest that the rate function of the hitting time offers a way to detect and quantify the presence of rare events in the dynamics. Such events indicate points where the local dimensions and the first hitting time take non-typical values. These points influence what happens in a finite region around their location. As one application, the authors study experimental data from climate dynamics, looking at atmospheric extreme events (such as extratropical storms or blocking episodes) which produce large excursions of the local dimensions. These, in turn, are associated with large deviations of hitting and return times in the proximity of special points in phase space. The researchers believe this approach offers statistical tools to investigate and interpret data coming from many fields, including turbulence, as well as in neuroscience and biology.
The paper is available at

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