mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, topological dynamics is a branch of the theory of

dynamical systems In mathematics, a dynamical system is a system in which a Function (mathematics), function describes the time dependence of a Point (geometry), point in an ambient space, such as in a parametric curve. Examples include the mathematical models ...

in which qualitative, asymptotic properties of dynamical systems are studied from the viewpoint of

general topology In mathematics, general topology (or point set topology) is the branch of topology that deals with the basic set-theoretic definitions and constructions used in topology. It is the foundation of most other branches of topology, including differ ...

Scope

The central object of study in topological dynamics is a topological dynamical system, i.e. a

topological space In mathematics, a topological space is, roughly speaking, a Geometry, geometrical space in which Closeness (mathematics), closeness is defined but cannot necessarily be measured by a numeric Distance (mathematics), distance. More specifically, a to ...

, together with a continuous transformation, a continuous flow, or more generally, a

semigroup In mathematics, a semigroup is an algebraic structure consisting of a set together with an associative internal binary operation on it. The binary operation of a semigroup is most often denoted multiplicatively (just notation, not necessarily th ...

of continuous transformations of that space. The origins of topological dynamics lie in the study of asymptotic properties of trajectories of systems of

autonomous In developmental psychology and moral, political, and bioethical philosophy, autonomy is the capacity to make an informed, uncoerced decision. Autonomous organizations or institutions are independent or self-governing. Autonomy can also be defi ...

ordinary differential equation In mathematics, an ordinary differential equation (ODE) is a differential equation (DE) dependent on only a single independent variable (mathematics), variable. As with any other DE, its unknown(s) consists of one (or more) Function (mathematic ...

s, in particular, the behavior of

limit set In mathematics, especially in the study of dynamical systems, a limit set is the state a dynamical system reaches after an infinite amount of time has passed, by either going forward or backwards in time. Limit sets are important because they c ...

s and various manifestations of "repetitiveness" of the motion, such as periodic trajectories, recurrence and minimality, stability, non-wandering points.

George Birkhoff George David Birkhoff (March21, 1884November12, 1944) was one of the top American mathematicians of his generation. He made valuable contributions to the theory of differential equations, dynamical systems, the four-color problem, the three-bo ...

is considered to be the founder of the field. A structure theorem for minimal distal flows proved by

Hillel Furstenberg Hillel "Harry" Furstenberg (; born September 29, 1935) is a German-born American-Israeli mathematician and professor emeritus at the Hebrew University of Jerusalem. He is a member of the Israel Academy of Sciences and Humanities and U.S. Natio ...

in the early 1960s inspired much work on classification of minimal flows. A lot of research in the 1970s and 1980s was devoted to topological dynamics of one-dimensional maps, in particular, piecewise linear self-maps of the interval and the circle. Unlike the theory of smooth dynamical systems, where the main object of study is a

smooth manifold In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One may ...

with a

diffeomorphism In mathematics, a diffeomorphism is an isomorphism of differentiable manifolds. It is an invertible function that maps one differentiable manifold to another such that both the function and its inverse are continuously differentiable. Definit ...

or a smooth flow,

phase space The phase space of a physical system is the set of all possible physical states of the system when described by a given parameterization. Each possible state corresponds uniquely to a point in the phase space. For mechanical systems, the p ...

s considered in topological dynamics are general

metric spaces In mathematics, a metric space is a set together with a notion of ''distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are a general setting for ...

(usually,

compact Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact, a type of agreement used by U.S. states * Blood compact, an ancient ritual of the Philippines * Compact government, a t ...

). This necessitates development of entirely different techniques but allows an extra degree of flexibility even in the smooth setting, because invariant subsets of a manifold are frequently very complicated topologically (cf

limit cycle In mathematics, in the study of dynamical systems with two-dimensional phase space, a limit cycle is a closed trajectory in phase space having the property that at least one other trajectory spirals into it either as time approaches infinity o ...

strange attractor In the mathematics, mathematical field of dynamical systems, an attractor is a set of states toward which a system tends to evolve, for a wide variety of starting conditions of the system. System values that get close enough to the attractor va ...

); additionally,

shift space Shift may refer to: Art, entertainment, and media Gaming * ''Shift'' (series), a 2008 online video game series by Armor Games * '' Need for Speed: Shift'', a 2009 racing video game ** '' Shift 2: Unleashed'', its 2011 sequel Literature * ''S ...

s arising via symbolic representations can be considered on an equal footing with more geometric actions. Topological dynamics has intimate connections with

ergodic theory Ergodic theory is a branch of mathematics that studies statistical properties of deterministic dynamical systems; it is the study of ergodicity. In this context, "statistical properties" refers to properties which are expressed through the behav ...

of dynamical systems, and many fundamental concepts of the latter have topological analogues (cf Kolmogorov–Sinai entropy and

topological entropy In mathematics, the topological entropy of a topological dynamical system is a nonnegative extended real number that is a measure of the complexity of the system. Topological entropy was first introduced in 1965 by Adler, Konheim and McAndrew. Th ...

References

* * * Robert Ellis, ''Lectures on topological dynamics''. W. A. Benjamin, Inc., New York 1969 * Walter Gottschalk, Gustav Hedlund, ''Topological dynamics''. American Mathematical Society Colloquium Publications, Vol. 36. American Mathematical Society, Providence, R. I., 1955 * J. de Vries, ''Elements of topological dynamics''. Mathematics and its Applications, 257. Kluwer Academic Publishers Group, Dordrecht, 1993 * Ethan Akin, ''The General Topology of Dynamical Systems'', AMS Bookstore, 2010, * J. de Vries, ''Topological Dynamical Systems: An Introduction to the Dynamics of Continuous Mappings'', De Gruyter Studies in Mathematics, 59, De Gruyter, Berlin, 2014, {{isbn, 978-3-1103-4073-0 * Jian Li and Xiang Dong Ye, ''Recent development of chaos theory in topological dynamics'', Acta Mathematica Sinica, English Series, 2016, Volume 32, Issue 1, pp. 83–114.

Scope

See also

References