Self-Organized Criticality

We extend a number of results from one-dimensional dynamics based on spectral properties of the Ruelle–Perron–Frobenius transfer operator to Anosov diffeomorphisms on compact manifolds. This allows us to develop a direct operator approach to study ergodic properties of these maps. In particular, we show that it is possible to define Banach spaces on which the transfer operator is quasi-compact. (Information on the existence of a Sinai–Ruelle–Bowen measure, its smoothness properties and statistical properties readily follow from such a result.) In dimension d = 2 we show that the transfer operator associated with smooth random perturbations of the map is close, in a proper sense, to the unperturbed transfer operator. This allows us to obtain easily very strong spectral stability results, which, in turn, imply spectral stability results for smooth deterministic perturbations as well. Finally, we are able to implement an Ulam-type finite rank approximation scheme thus reducing the study of the spectral properties of the transfer operator to a finite-dimensional problem.

https://iopscience.iop.org/article/10.1088/0951-7715/15/6/309/pdf

Ruelle-Perron-Frobenius spectrum for Anosov maps

We consider a class of discrete time random dynamical systems and establish the exponential convergence of its trajectories to a unique stationary measure. The result obtained applies, in particular, to the 2D Navier-Stokes system and multidimensional complex Ginzburg-Landau equation with random kick-force.

/pdf/a-coupling-approach-to-randomly-forced-nonlinear-pde-s-ii-2dfue03abj.pdf

A Coupling Approach to Randomly Forced Nonlinear PDE's. II

We consider a new class of non Markovian processes with a countable number of interacting components. At each time unit, each component can take two values, indicating if it has a spike or not at this precise moment. The system evolves as follows. For each component, the probability of having a spike at the next time unit depends on the entire time evolution of the system after the last spike time of the component. This class of systems extends in a non trivial way both the interacting particle systems, which are Markovian (Spitzer in Adv. Math. 5:246–290, 1970) and the stochastic chains with memory of variable length which have finite state space (Rissanen in IEEE Trans. Inf. Theory 29(5):656–664, 1983). These features make it suitable to describe the time evolution of biological neural systems. We construct a stationary version of the process by using a probabilistic tool which is a Kalikow-type decomposition either in random environment or in space-time. This construction implies uniqueness of the stationary process. Finally we consider the case where the interactions between components are given by a critical directed Erdos-Renyi-type random graph with a large but finite number of components. In this framework we obtain an explicit upper-bound for the correlation between successive inter-spike intervals which is compatible with previous empirical findings.

/pdf/infinite-systems-of-interacting-chains-with-memory-of-4jvc001s0s.pdf

Infinite Systems of Interacting Chains with Memory of Variable Length—A Stochastic Model for Biological Neural Nets

We present a perfect simulation algorithm for stationary processes indexed by $\mathbb{Z}$, with summable memory decay. Depending on the decay, we construct the process on finite or semi-infinite intervals, explicitly from an i.i.d. uniform sequence. Even though the process has infinite memory, its value at time 0 depends only on a finite, but random, number of these uniform variables. The algorithm is based on a recent regenerative construction of these measures by Ferrari, Maass, Martinez and Ney. As applications, we discuss the perfect simulation of binary autoregressions and Markov chains on the unit interval.

/pdf/processes-with-long-memory-regenerative-construction-and-31wau1mh0e.pdf

Processes with long memory: Regenerative construction and perfect simulation

We present an upper bound on the mixing rate of the equilibrium state of a dynamical system defined by the one-sided shift and a non Holder potential of summable variations. The bound follows from an estimation of the relaxation speed of chains with complete connections with summable decay, which is obtained via a explicit coupling between pairs of chains with different histories.

/pdf/decay-of-correlations-for-non-holderian-dynamics-a-coupling-541hcso9ou.pdf

Decay of Correlations for Non-Hölderian Dynamics. A Coupling Approach

We present an upper bound on the mixing rate of the equilibrium state of a dynamical systems defined by the one-sided shift and a non H\"{o}lder potential of summable variations. The bound follows from an estimation of the relaxation speed of chains with complete connections with summable decay, which is obtained via a explicit coupling between pairs of chains with different histories.

Decay of correlations for non H\"{o}lderian dynamics. A coupling approach

For T a continuous map from a compact metric space to itself and f a continuous function, we study the minimum of the integral of f with respect to the members of the family of invariant measures for T and in particular the rate at which this minimum is approached when the minimum is restricted to the family of invariant measures supported on periodic orbits of period at most N. We answer a question of Yuan and Hunt by demonstrating that the error of approximation decays faster than N−k for all k > 0 and show that this is sharp by giving examples for which the approximation error does not decay faster than this.

Rate of approximation of minimizing measures

Programa Iniciativa Cientifica
Milenio P01-005 and FONDECYT 1010447. This project was also partially supported by
the international cooperation programme ECOS-CONICYT C03-E03 and FONDECYT for
international cooperation agreement 7040166.

https://hal.archives-ouvertes.fr/hal-00224277/document

Necessary and sufficient conditions to be an eigenvalue for linearly recurrent dynamical Cantor systems

http://www.staff.science.uu.nl/%7Eferna107/papers/speed.pdf

Xavier Bressaud

Papers

Decay of Correlations for Non-Hölderian Dynamics. A Coupling Approach

Decay of correlations for non H\"{o}lderian dynamics. A coupling approach

Rate of approximation of minimizing measures

Necessary and sufficient conditions to be an eigenvalue for linearly recurrent dynamical Cantor systems

Speed of d-convergence for Markov approximations of chains with complete connections. A coupling approach☆