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Chaos and Ergodic Theory

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Mathematics of Complexity and Dynamical Systems

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Article Outline

Glossary

Definition of the Subject

Introduction

Picking an Invariant Probability Measure

Tractable Chaotic Dynamics

Statistical Properties

Orbit Complexity

Stability

Untreated Topics

Future Directions

Acknowledgments

Bibliography

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Abbreviations

Entropy, measure‐theoretic (or: metric entropy):

For an ergodic invariant probability measure μ, it is the smallest exponential growth rates of the number of orbit segments of given length, with respect to that length, after restriction to a set of positive measure. We denote it by \({h(T,\mu)}\). See Entropy in Ergodic Theory and Subsect. “Local Complexity” below.

Entropy, topological:

It is the exponential growth rates of the number of orbit segments of given length, with respect to that length. We denote it by \({h_{\operatorname{top}}(f)}\). See Entropy in Ergodic Theory and Subsect. “Local Complexity” below.

Ergodicity:

A  measure is ergodic with respect to a map T if given any measurable subset S which is invariant, i. e., such that \({T^{-1}S=S}\), either S or its complement has zero measure.

Hyperbolicity:

A  measure is hyperbolic in the sense of Pesin if at almost every point no Lyapunov exponent is zero. See Smooth Ergodic Theory.

Kolmogorov typicality:

A  property is typical in the sense of Kolmogorov for a topological space \( \mathcal F \) of parametrized families \( f=(f_t)_{t\in U}\), U being an open subset of \({\mathbb{R}}^d \) for some \( d\geq1 \), if it holds for f t for Lebesgue almost every t and topologically generic \({f \in \mathcal{F}}\).

Lyapunov exponents:

The Lyapunov exponents (Smooth Ergodic Theory) are the limits, when they exist, \( \lim_{n\to\infty} \frac1n \log\|(T^n)^{\prime}(x).v\| \) where \({x\in M}\) and v is a non zero tangent vector to M at x. The Lyapunov exponents of an ergodic measure is the set of Lyapunov exponents obtained at almost every point with respect to that measure for all non-zero tangent vectors.

Markov shift (topological, countable state):

It is the set of all infinite or bi‐infinite paths on some countable directed graph endowed with the left-shift, which just translates these sequences.

Maximum entropy measure:

It is a measure μ which maximizes the measured entropy and, by the variational principle, realized the topological entropy.

Physical measure:

It is a measure μ whose basin, \( \{x\in M\colon \forall \phi\colon M\to{\mathbb{R}}\text{ continuous }\lim_{n\to\infty} \frac1n\sum_{k=0}^{n-1}\phi(f^kx)=\int \phi\, \mskip2mu\mathrm{d}\mu\}\) has nonzero volume.

Prevalence:

A  property is prevalent in some complete metric, separable vector space X if it holds outside of a set N such that, for some Borel probability measure μ on X: \({\mu(A+v)=0}\) for all \({v\in X}\). See [76,141,239].

Sensitivity on initial conditions:

T has sensitivity to initial conditions on \({X^{\prime}\subset X}\) if there exists a constant \({\rho > 0}\) such that for every \({x\in X^{\prime}}\), there exists \({y\in X}\), arbitrarily close to x, and \({n\geq0}\) such that \( d(T^ny, T^nx) > \rho \).

Sinai–Ruelle–Bowen measures:

It is an invariant probability measure which is absolutely continuous along the unstable foliation (defined using the unstable manifolds of almost every \({x\in M}\), which are the sets \({W^u(x)}\), of points y such that \( \lim_{n\to\infty} \frac1n \log d(T^{-n}y,T^{-n}x)<0 \)).

Statistical stability:

T is statistically stable if the physical measures of nearby deterministic systems are arbitrarily close to the convex hull of the physical measures of T.

Stochastic stability:

T is stochastically stable if the invariant measures of the Markov chains obtained from T by adding a suitable, smooth noise with size \({\epsilon\to0}\) are arbitrarily close to the convex hull of the physical measures of T.

Structural stability:

T is structurally stable if any S close enough to T is topologically the same as T: there exists a homeomorphism \({h \colon M \to M}\) such that \({h\circ T=S\circ h}\) (orbits are sent to orbits).

Subshift of finite type:

It is a closed subset \({\Sigma_F}\) of \( \Sigma = \mathcal A^{\mathbb{Z}}\) or \({\Sigma=\mathcal A^{\mathbb{N}}}\) where \({\mathcal A}\) is a finite set satisfying: \( \Sigma_F = \{x\in\Sigma \colon \forall k < \ell \colon x_kx_{k+1}\dots x_\ell\notin F\}\) for some finite set F.

Topological genericity:

Let  X be a Baire space, e. g., a complete metric space. A property is (topologically) generic in a space X (or holds for the (topologically) generic element of X) if it holds on a nonmeager set (or set of second Baire category), i. e., on a dense \({G_\delta}\) subset.

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Acknowledgments

I am grateful for the advice and/or comments of the following colleagues: P. Collet, J.-R. Chazottes, and especially S. Ruette. I am also indebted to the anonymous referee.

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    Jérôme Buzzi

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Buzzi, J. (2012). Chaos and Ergodic Theory. In: Meyers, R. (eds) Mathematics of Complexity and Dynamical Systems. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-1806-1_6

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