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Geometrical definition

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The statement of the theorem

In the geometrical definition, a dynamical system is the tuple ⟨T,M,f⟩\langle {\mathcal {T}},{\mathcal {M}},f\rangle . T{\mathcal {T}} is the domain for time – there are many choices, usually the reals or the integers, possibly restricted to be non-negative. M{\mathcal {M}} is a manifold, i.e. locally a Banach space or Euclidean space, or in the discrete case a graph. f is an evolution rule t → f^{ t} (with t∈Tt\in {\mathcal {T}} ) such that f^{ t} is a diffeomorphism of the manifold to itself. So, f is a "smooth" mapping of the time-domain T{\mathcal {T}} into the space of diffeomorphisms of the manifold to itself. In other terms, f(t) is a diffeomorphism, for every time t in the domain T{\mathcal {T}} . A real dynamical system, real-time dynamical system, continuous time dynamical system, or flow is a tuple (T, M, Φ) with T an open interval in the real numbersR, M a manifold locally diffeomorphic to a Banach space, and Φ a continuous function. If Φ is continuously differentiable the system is called a differentiable dynamical system. If the manifold M is locally diffeomorphic to R^{n}, the dynamical system is finite-dimensional; if not, the dynamical system is infinite-dimensional. This does not assume a symplectic structure. When T is taken to be the reals, the dynamical system is called global or a flow; and if T is restricted to the non-negative reals, then the dynamical system is a semi-flow. A discrete dynamical system, discrete-time dynamical system is a tuple (T, M, Φ), where M is a manifold locally diffeomorphic to a Banach space, and Φ is a function. When T is taken to be the integers, it is a cascade or a map. If T is restricted to the non-negative integers the system is called a semi-cascade. A cellular automaton is a tuple (T, M, Φ), with T a lattice such as the integers or a higher-dimensional integer grid, M is a set of functions from an integer lattice (again, with one or more dimensions) to a finite set, and Φ a (locally defined) evolution function. As such cellular automata are dynamical systems. The lattice in M represents the "space" lattice, while the one in T represents the "time" lattice. Dynamical systems are usually defined over a single independent variable, thought of as time. A more general class of systems are defined over multiple independent variables and are therefore called multidimensional systems. Such systems are useful for modeling, for example, image processing. Given a global dynamical system (R, X, Φ) on a locally compact and Hausdorfftopological spaceX, it is often useful to study the continuous extension Φ* of Φ to the one-point compactificationX* of X. Even after losing the differential structure of the original system, there are compactness arguments to analyze the new system (R, X*, Φ*). In compact dynamical systems the limit set of any orbit is non-empty, compact and simply connected. - ^Galor, Oded (2010). Discrete Dynamical Systems. Springer.