Dynamical systems

By a dynamical system in general we mean a pair (X,T), where X is just a set (called a phase space), while T is a group or semigroup of selftransformations of X. In the classical case of a cascade these are iterates of a single transformation (then it is customary to use T to denote this transformation). Usually, the space X is endowed with some kind of structure, and the acting transformations respect this structure. The theory of dynamical systems is interdisciplinary, as it involves research methods from many other branches of mathematics. There has been an intense development of the theory over the past few decades and it grew into a science successfully competing for applications in the practical sciences with statistical and even with numerical techniques. See example.

And so, if X is a measure space (most often a probability space) then the transformations are assumed to be measurable and preserve the measure by preimage (sometimes one requires only that they are nonsingular). The branch investigating so defined dynamical systems is called ergodic theory.

If X is assumed to be a topological space (usually compact) than T is required to be continuous. These systems are subject of topological dynamics.

And finally, one can endow X with the structure of a C^r-differentiable manifold and consider a C^r-diffeomorfism T on X. This approach leads to the theory of smooth dynamical systems.

An important parameter of a dynamical system is its entropy. Most generally speaking, entropy measures complexity of the system or its degree of randomness. Entropy appears in all types of dynamical systems; in ergodic theory this will be measure-theoretic entropy of Kolmogorov-Sinai, in topological dynamics one uses topological entropy, e.g., as defined by Bowen. More detailed information about entropy can be found here.

Besides this classification there is also symbolic dynamics, in which X is a set of two-sided  infinite sequences {x_n: n = ...-2,-1, 0,1,2,...} or one-sided {x_n: n = 0,1,2,...}, taking values in a finite set (alphabet) A, while the transformation is the left shift: 

T({x_n}) = {x_n+1}.

Methods of symbolic dynamics are used in the investigations of all three above named types of dynamical systems and in the entropy theory. 

One considers a measure space (X, B, µ) and a measurable transformation T of X into X, preserving the measure by preimage. Such transformation is called ergodic if all T-invariant sets have measure zero or their complements have measure zero. Main result in this theory, the Birkhoff ergodic theorem asserts that if µ is a probability measure, T is ergodic, and f is a real (or complex) valued µ-absolutely integrable function on X, then the so-called Cesaro means 

[f(x)+f(Tx)+...+f(Tⁿx)]/n 

converge µ-almost everywhere to the integral of f with respect to µ (the time average equals the space average). 

A little earlier the Poincaré recurrence theorem has been proved: if T is a probability measure µ preserving transformation  and A is a measurable set of positive measure then µ-almost every point x of A has the property that Tⁿx belongs to A for some n > 0. This theorem has a physical interpretation seemingly in contradiction with the II law of thermodynamics: Consider two gasses in a container, which are initially not mixed. According to the II law, after some time they will get mixed together and never separate spontaneously. However, the Poincaré theorem says that there will be time when they nearly return to the initial state (where they were separated). This paradox can be explained as follows: the Poincaré return time is very large while the physical system is never perfectly isolated from random outer influences, hence it does not perfectly follow the same transformation (and the II law takes this into account). As result, before the return time arrives the system deviates so much from the perfect model that the recurrence to the initial state never occurs.

Mixing is a key notion in ergodic theory. A dynamical system is called mixing if for any two measurable sets A and B the measure of the intersection of T^-nA with B converges to the product µ(A)µ(B). Roughly this means that after a long time every set A will spread evenly over the entire space (its contribution in every set  B will be nearly proportional to the size of B). Every mixing system is ergodic, but e.g., the rotation of the unit circle by the angle irrational with respect to p is ergodic but not mixing (an arc remains an arc - it does not spread). 

The notions of ergodicity and mixing (and many others) mimic the behavior first observed  in physical systems. Modern methods of ergodic theory allow to describe and predict the future of a process without knowing the exact formulas for the trajectories of all involved particles.

In non-deterministic mechanics the state of a system observed at the present time (time zero) does not uniquely determine the states the system will assume in future instances of time, it only determines a probability distribution on such states in the future. This approach leads to stochastic processes. As it turns out, such processes can also be described in the language of ergodic theory. In order to achieve this one denotes by X not directly the set of all states only the set of all possible futures, i.e., sequences of states, with the present instance indexed as time zero. In the considerations with discrete time, in every consecutive instance the time scale is renumbered: new time zero is what was time 1, new time 1 is what was time 2, and so on. If the set of states A is assumed finite, this leads to a symbolic system: the space X is the set of all sequences {x_n: n = 0,1,2,...} with values in A and T is the shift transformation: 

T({x_n: n = 0,1,2,...}) = {x_n+1: n = 0,1,2,...}.

A classical example here is the random walk on a finite set of states A = {1,2,3,..., k} governed by a stochastic transition matrix P= [p_i,j]  (from state i the system passes to the state j with probability p_i,j).  The transition probabilities, along with some stationary initial distribution on the states, provides on X a shift-invariant measure; in this case that will be the measure of the corresponding Markov process.

Topological dynamics is the youngest of the three branches named in the introduction. Most often in this theory one considers a compact phase space X and a continuous map (or a homeomorphism) T from X to X. The physical interpretation of continuity is that two sufficiently close points travel similarly, at least within a specified bounded time period. A good intuition is provided by a flow of water in a river. If small initial distance between two points guarantees that they will remain close at all times, then we are dealing with a system with equicontinuous iterates. This is a very strong internal stability condition. Topological analogues of ergodicity are: transitivity - the existence of a point whose trajectory is dense in X, and minimality - the requirement that trajectories of all points are dense. 

A fundamental theorem connecting topological dynamics with ergodic theory is the Bogolubov-Krylov theorem stating that every compact topological dynamical system admits at least one T-invariant Borel-measurable probability measure. With respect to such measure, T becomes a measure preserving transformation and one can investigate its ergodicity or mixing properties. Note that even a minimal system may admit many invariant measures including nonergodic ones. The reversed connection between these theories is provided by the Jewett-Krieger theorem: every invertible measure preserving and ergodic transformation of a probability space is isomorphic (measure-theoretically) to a minimal topological system having only one invariant measure. 

Other notions frequently studied in topological dynamics are proximality, distality, equicontinuity of iterates or expansiveness. Two points are proximal if, during the evolution, they come arbitrarily near each-other. The system is called distal if it contains no proximal pairs. It is an easy exercise that a homeomorphism with equicontinuous iterates Tⁿ yields a distal system. The system is expansive if there exists a positive constant d such that for any two distinct points there is time when they appear at least d apart.

Symbolic systems can also be viewed as topological. By the full shift over a finite alphabet A (considered a discrete topological space) we mean the set X of all (two-sided or on-sided) A-valued sequences endowed with the Tichonov topology (which in this case is zero-dimensional), and the shift transformation T{x_n}  = {x_n+1}. This is clearly a continuous transformation and in the two-sided case  (n ranges over Z) even a homeomorphism. Generally, by a symbolic system we mean the action of the T on any closed shift-invariant subset Y of X. All symbolic systems are expansive (hence, unless Y is finite, do not have equicontinuous iterates). Hedlund theorem asserts that every expansive system on a compact zero-dimensional space is equivalent to a symbolic system.

An abstract class of topological dynamical systems are rotations of topological groups (not necessarily commutative). In the group G we select an element  g₀ and let T be the left multiplication in G by g₀, namely

T(g) = gg₀.

By the axioms of topological groups,  such T is continuous. A classical example is the irrational rotation of the unit circle. This system is minimal and has equicontinuous iterates. One of the important theorems in topological dynamics, the Halmos - von Neumann theorem says that every minimal compact system with equicontinuous iterates is a group rotation. As an application of the dynamics of group rotations we obtain an immediate proof that numbers of the form sin(n) are dense in [-1,1] (1 is irrational with respect to p hence the trajectory of the complex number 1 is dense in the circle. The statement now follows by projection onto the imaginary axis).

Historically, this is the oldest branch of the theory of dynamical systems. It emerged from physics. A smooth system may be used to describe the evolution of a physical system constrained by a system of (often partial) differential equations. A classical example is the system of n planets with different masses, attracting each-other with gravity forces counterbalanced by centrifugal forces in the circulation of ones about others. The states (points of the phase space X) are vectors

 (x₁,y₁,z₁,u₁,v₁,w₁,  x₂,y₂,z₂,u₂,v₂,w₂ , ...,  x_n,y_n,z_n,u_n,v_n,w_n), 

where every six coordinates identifies the position and velocity of one planet. In classical mechanics a current state determines completely the future of the system. The trajectories of the planets are solutions of certain differential equations, and the obtained system is smooth. Solutions of systems of differential equations lead to flows rather than cascades. A flow is a dynamical system with the action of transformations indexed by real numbers, not only integers (system with continuous time). An example is a geodesic flow. It is defined on a direction field of a differentiable manifold and the trajectories follow geodesic curves. Best understood are geodesic flows on surfaces with negative curvature. 

Generally speaking, a smooth dynamical system is an action of a C^r-diffeomorphism (with r natural or infinite), on a differentiable manifold, typically compact, connected and without boundary. Since a diffeomorphism is continuous, one deals with a topological dynamical system and it makes sense to ask about transitivity, minimality or topological entropy. In the compact case one also has invariant measures, which makes all ergodic-theoretic properties available. But the differentiable structure allows a much deeper study of the behavior of the system, for instance one can measure the degree of expansion or contraction in certain directions near some points. This leads to the key notion of hyperbolicity. A fixed (or periodic) point x is hyperbolic if, in the local system of coordinates, T behaves like a linear transformation without unimodular (complex) eigenvalues. In the neighborhood of such point one can isolate two invariant locally complementary submanifolds: stable W ^- and unstable W⁺, such that consecutive images of every point in W⁺ exponentially approach x, and similarly behave preimages of points in W ^-. A closed T-invariant subset L of the manifold X is called hyperbolic if the tangent bundle restricted to L decomposes as a simple sum of two invariant subbundles E⁺ and E^- such that the derivative dT restricted to the first is exponentially contracting and it is exponentially expanding on the latter. The system (X,T) is called Anosov if the whole manifold X happens to be a hyperbolic set. Anosov systems are probably the best understood  smooth systems.

With the manifold X fixed, there is a natural C^r-topology in the family of all C^r-diffeomorphisms on X. This enables one to consider stability of certain properties of the systems. A property is called stable if it is possessed by all maps S from a sufficiently small neighborhood of T in this topology. A diffeomorphism T is called structurally stable if all diffeomorphisms S from some neighborhood of T are topologically conjugate to T. Stability has an extremely important interpretation in physics: for example, if a physical system is described by a structurally stable smooth dynamical system then if a small perturbation occurs, it will not essentially change the system's behavior even over a long run. It is worth mentioning that already the XVIIIth century astronomers have wondered whether the solar system is stable (it would be very reassuring to know such thing). An important result in stability theory, proved by Anosov, is that every Anosov system is structurally stable.