TITLE

# Complexity for Extended Dynamical Systems

AUTHOR(S)
Bonanno, Claudio; Collet, Pierre
PUB. DATE
October 2007
SOURCE
Communications in Mathematical Physics;Oct2007, Vol. 275 Issue 3, p721
SOURCE TYPE
DOC. TYPE
Article
ABSTRACT
We consider dynamical systems for which the spatial extension plays an important role. For these systems, the notions of attractor, Ïµ-entropy and topological entropy per unit time and volume have been introduced previously. In this paper we use the notion of Kolmogorov complexity to introduce, for extended dynamical systems, a notion of complexity per unit time and volume which plays the same role as the metric entropy for classical dynamical systems. We introduce this notion as an almost sure limit on orbits of the system. Moreover we prove a kind of variational principle for this complexity.
ACCESSION #
26430060

## Related Articles

• A variational principle for topological pressure for certain non-compact sets. Thompson, Daniel // Journal of the London Mathematical Society;Dec2009, Vol. 80 Issue 3, p585

Let (X, d) be a compact metric space, let f:X â†¦ X be a continuous map with the specification property and let Ï•: X â†¦ â„ be a continuous function. We prove a variational principle for topological pressure (in the sense of Pesin and Pitskel) for non-compact sets of the form...

• ENTROPY OPERATOR FOR CONTINUOUS DYNAMICAL SYSTEMS OF FINITE TOPOLOGICAL ENTROPY. RAHIMI, MEHDI; RIAZI, ABDOLHAMID // Bulletin of the Iranian Mathematical Society;Dec2012, Vol. 38 Issue 4, p883

In this paper we introduce the concept of entropy operator for a continuous system of finite topological entropy. It is shown that it generates the Kolmogorov entropy as a special case. If Ï† is invertible then the entropy operator is bounded by the topological entropy of Ï† as its norm.

• Pointwise Variation Growth and Entropy of the Descartes Product of a Few of Interval Maps. Risong Li; Zengxiong Cheng // Pure Mathematics;Oct2011, Vol. 1 Issue 3, p184

In this paper, the definition of pointwise variation growth of interval maps was extended to continuous self-maps on k-dimensional space I1 x I2 xï¿½x Ik, where Ii is a closed interval. Let fi : Ii & larr; Ii be a continuous map and the total variation Due to image rights restrictions,...

• On the relation between topological entropy and entropy dimension. Saltykov, P. S. // Mathematical Notes;Jun2009, Vol. 86 Issue 1/2, p255

For the Lipschitz mapping of a metric compact set into itself, there is a classical upper bound on topological entropy, namely, the product of the entropy dimension of the compact set by the logarithm of the Lipschitz constant. The Ghys conjecture is that, by varying the metric, one can...

• Finite- and infinite-dimensional attractors for porous media equations. M. Efendiev; S. Zelik // Proceedings of the London Mathematical Society;Jan2008, Vol. 96 Issue 1, p51

The fractal dimension of the global attractors of porous media equations in bounded domains is studied. The conditions which guarantee this attractor to be finite dimensional are found and the examples of infinite-dimensional attractors that do not satisfy these conditions are constructed. The...

• Metric Entropy of Convex Hulls in Hilbert Spaces. Carl, Bernd // Bulletin of the London Mathematical Society;1997, Vol. 29 Issue 4, p452

We show in this note the following statement which is an improvement over a result of R. M. Dudley and which is also of independent interest. Let X be a set of a Hilbert space with the property that there are constants Ï, Ïƒ>0, and for each nâˆˆ N, the set X can be covered by at most n...

• Computing Topological Entropy in a Space of Quartic Polynomials. Radulescu, Anca // Journal of Statistical Physics;Jan2008, Vol. 130 Issue 2, p373

This paper adds a computational approach to a previous theoretical result illustrating how the complexity of a simple dynamical system evolves under deformations. The algorithm targets topological entropy in the 2-dimensional family P Q of compositions of two logistic maps. Estimation of the...

• Entropy and collapsing of compact complex surfaces G. P. Paternain was partially supported by CIMAT, Guanajuato, MÃ©xico. J. Petean is supported by grant 37558-E of CONACYT.. Gabriel P. Paternain; Jimmy Petean // Proceedings of the London Mathematical Society;Nov2004, Vol. 89 Issue 3, p763

We study the problem of existence of $\mathcal{F}$-structures (in the sense of Cheeger and Gromov, but not necessarily polarized) on compact complex surfaces. We give a complete classification of compact complex surfaces of KÃ¡hler type admitting $\mathcal{F}$-structures. In the...

• Localizing Entropies via an Open Cover. Kesong Yan; Fanping Zeng; Qi Wang // Journal of Concrete & Applicable Mathematics;Jan2008, Vol. 6 Issue 1, p209

For a given topological dynamical system (X, T), we introduce and study some entropies of open covers. The main result as follows: (1)the inequality hp(T, U) â‰¤ hm(T, U) â‰¤ htop(T, U) â‰¤ hm(T, U) + hb(T), relating pointwise pre-image entropies of open covers hp(T, U), hm(T, U),...

Share