Orbit complexity and data compression
Stefano Galatolo
Discrete & Continuous Dynamical Systems - A 2001, 7(3): 477-486 doi: 10.3934/dcds.2001.7.477
We consider data compression algorithms as a tool to get an approximate measure for the quantity of information contained in a string. By this it is possible to give a notion of orbit complexity for topological dynamical systems. In compact ergodic dynamical systems, entropy is almost everywhere equal to orbit complexity. The use of compression algorithms allows a direct estimation of the information content of the orbits.
keywords: entropy Topological dynamical systems information orbit complexity. coding
Long hitting time, slow decay of correlations and arithmetical properties
Stefano Galatolo Pietro Peterlongo
Discrete & Continuous Dynamical Systems - A 2010, 27(1): 185-204 doi: 10.3934/dcds.2010.27.185
Let $\tau _r(x,x_0)$ be the time needed for a point $x$ to enter for the first time in a ball $B_r(x_0)$ centered in $x_0$, with small radius $r$. We construct a class of translations on the two torus having particular arithmetic properties (Liouville components with intertwined denominators of convergents) not satisfying a logarithm law, i.e. such that for typical $x,x_0$

liminfr → 0 $ \frac{\log \tau _r(x,x_0)}{-\log r} = \infty.$

   By considering a suitable reparametrization of the flow generated by a suspension of this translation, using a previous construction by Fayad, we show the existence of a mixing system on three torus having the same properties. The speed of mixing of this example must be subpolynomial, because we also show that: in a system having polynomial decay of correlations, the limsupr → 0 of the above ratio of logarithms (which is also called the upper hitting time indicator) is bounded from above by a function of the local dimension and the speed of correlation decay.
   More generally, this shows that reparametrizations of torus translations having a Liouville component cannot be polynomially mixing.

keywords: Arithmetical properties. Hitting time Dimension Decay of correlations
Global and local complexity in weakly chaotic dynamical systems
Stefano Galatolo
Discrete & Continuous Dynamical Systems - A 2003, 9(6): 1607-1624 doi: 10.3934/dcds.2003.9.1607
The generalized complexity of an orbit of a dynamical system is defined by the asymptotic behavior of the information that is necessary to describe $n$ steps of the orbit as $n$ increases. This local complexity indicator is also invariant up to topological conjugation and is suited for the study of $0$-entropy dynamical systems. First, we state a criterion to find systems with "non trivial" orbit complexity. Then, we consider also a global indicator of the complexity of the system. This global indicator generalizes the topological entropy, having non trivial values for systems were the number of essentially different orbits increases less than exponentially. Then we prove that if the system is constructive ( if the map can be defined up to any given accuracy by some algorithm) the orbit complexity is everywhere less or equal than the generalized topological entropy. Conversely there are compact non constructive examples where the inequality is reversed, suggesting that the notion of constructive map comes out naturally in this kind of complexity questions.
keywords: generalized topological entropy computable analisys. Weak chaos generalized orbit complexity
Dynamical systems and computable information
Vieri Benci C. Bonanno Stefano Galatolo G. Menconi M. Virgilio
Discrete & Continuous Dynamical Systems - B 2004, 4(4): 935-960 doi: 10.3934/dcdsb.2004.4.935
We present some new results that relate information to chaotic dynamics. In our approach the quantity of information is measured by the Algorithmic Information Content (Kolmogorov complexity) or by a sort of computable version of it (Computable Information Content) in which the information is measured by using a suitable universal data compression algorithm. We apply these notions to the study of dynamical systems by considering the asymptotic behavior of the quantity of information necessary to describe their orbits. When a system is ergodic, this method provides an indicator that equals the Kolmogorov-Sinai entropy almost everywhere. Moreover, if the entropy is null, our method gives new indicators that measure the unpredictability of the system and allows various kind of weak chaos to be classified. Actually, this is the main motivation of this work. The behavior of a 0-entropy dynamical system is far to be completely predictable except that in particular cases. In fact there are 0-entropy systems that exhibit a sort of weak chaos, where the information necessary to describe the orbit behavior increases with time more than logarithmically (periodic case) even if less than linearly (positive entropy case). Also, we believe that the above method is useful to classify 0-entropy time series. To support this point of view, we show some theoretical and experimental results in specific cases.
keywords: data compression. weak chaos Information
An elementary way to rigorously estimate convergence to equilibrium and escape rates
Stefano Galatolo Isaia Nisoli Benoît Saussol
Journal of Computational Dynamics 2015, 2(1): 51-64 doi: 10.3934/jcd.2015.2.51
We show an elementary method to obtain (finite time and asymptotic) computer assisted explicit upper bounds on convergence to equilibrium (decay of correlations) and escape rates for systems satisfying a Lasota Yorke inequality. The bounds are deduced from the ones of suitable approximations of the system's transfer operator. We also present some rigorous experiments on some nontrivial example.
keywords: interval arithmetics Ulam method. Rigorous computation decay of correlation convergence to equilibrium
Dynamics and abstract computability: Computing invariant measures
Stefano Galatolo Mathieu Hoyrup Cristóbal Rojas
Discrete & Continuous Dynamical Systems - A 2011, 29(1): 193-212 doi: 10.3934/dcds.2011.29.193
We consider the question of computing invariant measures from an abstract point of view. Here, computing a measure means finding an algorithm which can output descriptions of the measure up to any precision. We work in a general framework (computable metric spaces) where this problem can be posed precisely. We will find invariant measures as fixed points of the transfer operator. In this case, a general result ensures the computability of isolated fixed points of a computable map.
     We give general conditions under which the transfer operator is computable on a suitable set. This implies the computability of many "regular enough" invariant measures and among them many physical measures.
     On the other hand, not all computable dynamical systems have a computable invariant measure. We exhibit two examples of computable dynamics, one having a physical measure which is not computable and one for which no invariant measure is computable, showing some subtlety in this kind of problems.
keywords: Physical Measure Computable Analysis Computation Dynamical System.

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