# Generalized dimensions, large deviations and the distribution of rare events

@article{Caby2018GeneralizedDL, title={Generalized dimensions, large deviations and the distribution of rare events}, author={Theophile Caby and Davide Faranda and Giorgio Mantica and Sandro Vaienti and Pascal Yiou}, journal={arXiv: Dynamical Systems}, year={2018} }

Generalized dimensions of multifractal measures are usually seen as static objects, related to the scaling properties of suitable partition functions, or moments of measures of cells. When these measures are invariant for the flow of a chaotic dynamical system, generalized dimensions take on a dynamical meaning, as they provide the rate function for the large deviations of the first hitting time, which is the (average) time required to connect any two different regions in phase space. We prove… Expand

#### Figures and Tables from this paper

#### 15 Citations

Sampling Hyperspheres via Extreme Value Theory: Implications for Measuring Attractor Dimensions

- Mathematics
- Journal of Statistical Physics
- 2020

The attractor Hausdorff dimension is an important quantity bridging information theory and dynamical systems, as it is related to the number of effective degrees of freedom of the underlying… Expand

On the Computation of the Extremal Index for Time Series

- Mathematics, Physics
- Journal of Statistical Physics
- 2019

The extremal index is a quantity introduced in extreme value theory to measure the presence of clusters of exceedances. In the dynamical systems framework, it provides important information about the… Expand

Probability distributions for analog-to-target distances

- Mathematics
- Journal of the Atmospheric Sciences
- 2021

Some properties of chaotic dynamical systems can be probed through features of recurrences, also called analogs. In practice, analogs are nearest neighbours of the state of a system, taken from a… Expand

Rare Events for Cantor Target Sets

- Mathematics
- 2019

We study the existence of limiting laws of rare events corresponding to the entrance of the orbits on certain target sets in the phase space. The limiting laws are obtained when the target sets… Expand

Applications of large deviation theory in geophysical fluid dynamics and climate science

- Physics
- 2021

The climate is a complex, chaotic system with many degrees of freedom. Attaining a deeper level of understanding of climate dynamics is an urgent scientific challenge, given the evolving climate… Expand

Matching of observations

- Mathematics
- 2021

We study the statistical distribution of the closest encounter between observations computed along different trajectories of a mixing dynamical system. At the limit of large trajectories, the… Expand

Shortest distance between multiple orbits and generalized fractal dimensions

- Mathematics, Computer Science
- ArXiv
- 2019

This work considers rapidly mixing dynamical systems and links the decay of the shortest distance between multiple orbits with the generalized fractal dimension, and gets a relation between the longest common substring between multiple sequences and the generalized R\'enyi entropy. Expand

Using local dynamics to explain analog forecasting of chaotic systems

- Computer Science, Physics
- 2020

This study investigates the properties of different analog forecasting strategies by taking local approximations of the system's dynamics and finds that analog forecasting performances are highly linked to the local Jacobian matrix of the flow map, and that analog forecasts combined with linear regression allows to capture projections of this Jacobian Matrix. Expand

Extreme Value Theory with Spectral Techniques: application to a simple attractor

- Mathematics, Physics
- 2020

We give a brief account of application of extreme value theory in dynamical systems by using perturbation techniques associated to the transfer operator. We will apply it to the baker's map and we… Expand

Point Processes of Non stationary Sequences Generated by Sequential and Random Dynamical Systems

- Computer Science, Mathematics
- 2019

We give general sufficient conditions to prove the convergence of marked point processes that keep record of the occurrence of rare events and of their impact for non-autonomous dynamical systems. We… Expand

#### References

SHOWING 1-10 OF 74 REFERENCES

The Global Statistics of Return Times: Return Time Dimensions Versus Generalized Measure Dimensions

- Mathematics, Physics
- 2010

We investigate return times in dynamical systems, i.e. the time required by a trajectory to complete a return journey to a neighborhood of the initial position. In particular, we study the relations… Expand

Statistical description of chaotic attractors: The dimension function

- Mathematics
- 1985

A method for the investigation of fractal attractors is developed, based on statistical properties of the distributionP(δ, n) of nearest-neighbor distancesδ between points on the attractor. A… Expand

Erratum: Fractal measures and their singularities: The characterization of strange sets

- Physics, Medicine
- Physical review. A, General physics
- 1986

A description of normalized distributions (measures) lying upon possibly fractal sets; for example those arising in dynamical systems theory, focusing upon the scaling properties of such measures, which are characterized by two indices: \ensuremath{\alpha}, which determines the strength of their singularities; and f, which describes how densely they are distributed. Expand

Fractal dimensions and (a) spectrum of the Hnon attractor

- Physics
- 1987

Abstract We measure the generalized fractal dimensions D q ( q ⩾0) of the Henon attractor by the box counting and spatial correlation methods. The technique of virtual memory is exploited to handle… Expand

Correlation dimension and phase space contraction via extreme value theory.

- Mathematics, Physics
- Chaos
- 2018

It is shown how to obtain theoretical and numerical estimates of correlation dimension and phase space contraction by using the extreme value theory, and the extremal index is associated with the rate ofphase space contraction for backward iteration. Expand

An Improved Multifractal Formalism and Self Similar Measures

- Mathematics
- 1995

Abstract To characterize the geometry of a measure, its generalized dimensions dq have been introduced recently. The mathematically precise definition given by Falconer ["Fractal Geometry," 1990]… Expand

Extremes and Recurrence in Dynamical Systems

- Mathematics, Physics
- 2016

This book provides a comprehensive introduction for the study of extreme events in the context of dynamical systems. The introduction provides a broad overview of the interdisciplinary research area… Expand

Entropy estimation and fluctuations of Hitting and Recurrence Times for Gibbsian sources

- Mathematics
- 2004

Motivated by entropy estimation from chaotic time series, we provide a comprehensive analysis of hitting times of cylinder sets in the setting of Gibbsian sources. We prove two strong approximation… Expand

Multifractal properties of return time statistics.

- Medicine, Physics
- Physical review letters
- 2002

The global statistics of the return times of a dynamical system can be described by a new spectrum of generalized dimensions, and the difference between the two corresponding sets of dimensions is established. Expand

Dynamical proxies of North Atlantic predictability and extremes

- Mathematics, Medicine
- Scientific reports
- 2017

Instantaneous properties of the attractor provide an efficient way of evaluating and informing operational weather forecasts and show the existence of a significant correlation between the time series of instantaneous stability and dimension and the mean spread of sea-level pressure fields in an operational ensemble weather forecast. Expand