List of Dynamical Systems

Introduction

This article lists the dynamical systems the corpus introduces. A dynamical system is a space with a transformation or a flow that preserves a structure, and a row below names one system, records its entropy, its ergodicity and its mixing properties, and points to the article that introduces it. Every row points to an article; this article introduces nothing and proves nothing. The systems are grouped by the layer that introduces them: the rotations and translations, the expanding maps and the symbolic systems, the hyperbolic systems, the systems of geometry and mathematical physics, and the statistical invariants — entropy, Lyapunov exponent and Bowen–Ruelle measure — that the theory computes. The rows that record a failure — a rotation that is ergodic but not mixing, a toral automorphism that is not hyperbolic, an integrable system whose entropy vanishes — stand beside the systems as non-examples.

Rotations, Translations and Equidistribution

Object	Its entropy, ergodicity and mixing	Introduced in
the irrational rotation $x \mapsto x+\alpha$	entropy $0$; ergodic, and not weakly mixing: the Koopman operator has the full circle of eigenvalues	Ergodic Theory; Equidistribution
the rational rotation $x \mapsto x+p/q$	entropy $0$; not ergodic, its orbits being the cosets of $\frac1q\mathbb{Z}/\mathbb{Z}$	Ergodic Theory
the rotation on a torus, and the Kronecker flow	entropy $0$; ergodic for independent irrational frequencies, and not mixing	Ergodic Theory of Group Actions; Equidistribution
the doubling map $x \mapsto 2x \bmod 1$	entropy $\log2$; ergodic and mixing; the one-sided binary shift	Ergodic Theory
the Gauss map $x \mapsto 1/x \bmod 1$	entropy $\pi^2/(6\log2)$; ergodic and mixing for the Gauss measure; the generator of the continued fraction	Ergodic Theory; Diophantine Approximation and Continued Fractions
a general expanding map of the circle and of an interval	entropy $\int\log\lvert T'\rvert\,d\mu$; ergodic and mixing for an absolutely continuous invariant measure	Smooth Dynamical Systems

Symbolic and Markov Systems

Object	Its entropy, ergodicity and mixing	Introduced in
the full shift on $k$ symbols	topological entropy $\log k$; the Bernoulli shift with the uniform measure is ergodic and mixing	Symbolic Dynamics
the Bernoulli shift $(\sigma,\mu)$	metric entropy $H(p) = \sum_ip_i\log(1/p_i)$; ergodic and mixing; classified by Ornstein's theorem	Ergodic Theory; Symbolic Dynamics
a subshift of finite type, and a topological Markov chain	topological entropy $\log\rho(A)$ with $\rho(A)$ the spectral radius of the transition matrix; the Markov measure is ergodic and mixing when the matrix is irreducible and aperiodic	Symbolic Dynamics
a sofic shift	topological entropy $\log\rho(A)$ of the underlying graph; the measure-theoretic properties as in the finite-type case	Symbolic Dynamics; Dynamics and Number Theory
a suspension flow over a symbolic system	entropy of the flow $h_\mu(T)/\int\tau\,d\mu$ with $\tau$ the roof function	Hyperbolic Dynamics and Anosov Systems
a Markov partition of a hyperbolic system	the symbolic model conjugate to the system off a measure-zero set	Hyperbolic Dynamics and Anosov Systems

Hyperbolic Systems

Object	Its entropy, ergodicity and mixing	Introduced in
a hyperbolic toral automorphism $T_A$, $\lvert\operatorname{tr}A\rvert>2$	entropy $\log\lvert\lambda\rvert$ for the larger eigenvalue; ergodic, mixing, and the prototype of hyperbolicity	Ergodic Theory; Homogeneous Dynamics
an Anosov diffeomorphism	topological entropy as the maximum and metric entropy given by Pesin's formula; ergodic and mixing in the volume case	Hyperbolic Dynamics and Anosov Systems
an Anosov flow	entropy from the Lyapunov exponent; the Bowen–Ruelle measure is the equilibrium state	Hyperbolic Dynamics and Anosov Systems
an Axiom A diffeomorphism and its spectral decomposition	the decomposition into finitely many topologically transitive basic sets, each with a Markov partition	Hyperbolic Dynamics and Anosov Systems; Smooth Dynamical Systems
the Smale horseshoe	topological entropy $\log2$ on the invariant Cantor set; the model of chaotic behaviour	Chaos and Strange Attractors; Bifurcation Theory
a hyperbolic set with the shadowing and specification properties	the pseudo-orbit is shadowed by a true orbit; specification gives the entropy and the periodic-point statistics	Hyperbolic Dynamics and Anosov Systems

The Systems of Geometry and Physics

Object	Its entropy, ergodicity and mixing	Introduced in
the geodesic flow of a compact hyperbolic surface	entropy equal to the topological entropy of the surface, positive; ergodic and mixing for the Liouville measure	The Geodesic Flow; Hyperbolic Dynamics and Anosov Systems
a homogeneous flow and a unipotent flow	the Ratner classification of closures and equidistribution; equidistributed in the homogeneous space	Homogeneous Dynamics; Ratner's Theorems
a flow of an ergodic action of a Lie group	ergodic and mixing for the Haar measure under the Howe–Moore property	Ergodic Theory of Group Actions
a Hamiltonian system, and a Lagrangian system	entropy $0$ in the integrable case; the symplectic form and the Euler–Lagrange flow in general	Lagrangian and Hamiltonian Systems
an integrable system and the soliton equations	entropy $0$; the invariant tori of Liouville and the inverse-scattering solution	Integrable Systems; Soliton Theory
the Lorenz system	positive Lyapunov exponents and a strange attractor; the numerical prototype of chaos	Chaos and Strange Attractors
the Hénon map	a strange attractor of non-integer dimension; the Kaplan–Yorke formula	Chaos and Strange Attractors
the logistic family $x \mapsto \mu x(1-x)$	period doubling, the Feigenbaum cascade and the chaotic window; the entropy grows with $\mu$	Bifurcation Theory; Chaos and Strange Attractors
a random dynamical system	the multiplicative ergodic theorem gives the Lyapunov spectrum; the invariant measure is random	Random Dynamical Systems
a stochastic flow and its generator	the Markov semigroup and the Fokker–Planck equation; the invariant measure and its rate of convergence	Stochastic Differential Equations

The Statistical Invariants

Object	What it measures	Introduced in
the Kolmogorov–Sinai entropy	the exponential rate of information, $h_\mu(T)$; an isomorphism invariant, computed by the Kolmogorov–Sinai theorem	Ergodic Theory
the topological entropy	the exponential growth of the number of distinguishable orbit segments; the maximum of the metric entropies	Topological Dynamics; Symbolic Dynamics
ergodicity	the absence of nontrivial invariant sets; the identity of time and space averages	Ergodic Theory
mixing, weak mixing and the K-property	the decay of correlations, and the hierarchy of mixing notions; the K-property implies mixing, which implies ergodicity	Ergodic Theory
the Lyapunov exponents	the exponential rates of separation of nearby orbits; the Oseledets theorem gives their existence	Hyperbolic Dynamics and Anosov Systems
Pesin's formula	the identity of the metric entropy with the total positive Lyapunov exponent	Ergodic Theory; Hyperbolic Dynamics and Anosov Systems
the Bowen–Ruelle measure	the invariant measure that is the equilibrium state for the geometric potential, the physical measure of an attractor	Hyperbolic Dynamics and Anosov Systems; The Geodesic Flow
the invariant measure and the ergodic decomposition	the measure preserved by the system, decomposed into ergodic components	Ergodic Theory
the Poincaré recurrence theorem	the almost-sure return of the orbit to a set of positive measure	Ergodic Theory; Topological Dynamics

Systems and Invariants That Fail a Property

Object	The property that fails	Introduced in
the irrational rotation	is ergodic but not weakly mixing: the Koopman operator has eigenvalues of modulus one	Ergodic Theory
the rational rotation	is not ergodic; the invariant sets are the unions of cosets	Ergodic Theory
the identity map	has entropy $0$ and is not ergodic when the space is not a single atom	Ergodic Theory
an integrable Hamiltonian system	has zero entropy, so it is not chaotic; the KAM tori obstruct ergodicity	Lagrangian and Hamiltonian Systems; Integrable Systems
a non-hyperbolic toral automorphism, $\lvert\operatorname{tr}A\rvert \leq 2$	has no hyperbolic splitting and entropy $0$; the elliptic and parabolic cases	Ergodic Theory; Homogeneous Dynamics
a system with a rigid factor	is ergodic but not mixing	Ergodic Theory
a measure-preserving transformation with an invariant set of intermediate measure	is not ergodic, however large the set	Ergodic Theory
two Bernoulli shifts of different entropy	are not isomorphic; Ornstein's theorem separates them	Ergodic Theory
a system whose only invariant measure is supported on a periodic orbit	has vanishing metric entropy and no chaotic statistics	Chaos and Strange Attractors; Topological Dynamics

Summary

This list gathers the dynamical systems of the corpus: the circle rotations and translations, the doubling and Gauss maps, the Bernoulli and Markov shifts, the hyperbolic toral automorphisms, Anosov and Axiom A systems and the horseshoe, the geodesic and homogeneous flows, the Hamiltonian, integrable, Lorenz, Hénon and logistic systems, the random and stochastic systems, and the invariants — Kolmogorov–Sinai and topological entropy, ergodicity, mixing, the Lyapunov exponents, the Bowen–Ruelle measure and the Poincaré recurrence theorem. The closing table records the systems and invariants that fail a property of the theory.

Summary of Notation

The objects are named by their standard symbols; the tables use the following.

Symbol	Meaning
$T$, $\sigma$	a transformation, the shift
$\mu$, $h_\mu(T)$	an invariant measure, the Kolmogorov–Sinai entropy
$T_A$	the toral automorphism induced by $A \in SL_n(\mathbb Z)$
$\lambda$, $\chi_i$	an eigenvalue, a Lyapunov exponent
$H(p)$, $\rho(A)$	the entropy of a probability vector, the spectral radius of a transition matrix
$h_{\mathrm{top}}$	topological entropy