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Symbolic dynamics originated as a method to study general dynamical systems; now its techniques and ideas have found significant applications in data storage and transmission, linear algebra, the motions of the planets and many other areas [citation needed]. The distinct feature in symbolic dynamics is that time is measured in discrete intervals.
One of Hedlund's early results was an important theorem about the ergodicity of geodesic flows. [7] He also made significant contributions to symbolic dynamics, whose origins as a field of modern mathematics can be traced to a 1944 paper of Hedlund, and to topological dynamics.
Bowen: "Symbolic Dynamics for Hyperbolic Flows" in Proceedings of the International Congress of Mathematicians (Vancouver, 1974), pp. 299–302. Bowen: Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms. (Lecture Notes in Mathematics, no. 470: A. Dold and B. Eckmann, editors). Springer-Verlag (Heidelberg, 1975), 108 pp.
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The horseshoe map was designed to reproduce the chaotic dynamics of a flow in the neighborhood of a given periodic orbit. The neighborhood is chosen to be a small disk perpendicular to the orbit . As the system evolves, points in this disk remain close to the given periodic orbit, tracing out orbits that eventually intersect the disk once again.
Adler studies dynamical systems, ergodic theory, symbolic and topological dynamics and coding theory. The road coloring problem that was solved by Avraham Trakhtman in 2007 came from him, along with L. W. Goodwyn and Benjamin Weiss. He was a fellow of the American Mathematical Society. [4]
A Markov partition in mathematics is a tool used in dynamical systems theory, allowing the methods of symbolic dynamics to be applied to the study of hyperbolic dynamics.By using a Markov partition, the system can be made to resemble a discrete-time Markov process, with the long-term dynamical characteristics of the system represented as a Markov shift.
Berthé's research spans the area of symbolic dynamics, combinatorics on words, numeration systems and discrete geometry. She has recently made significant process in the study of S-adic dynamical systems, and also of continued fractions in higher dimensions. [5] [6] [7] [8]