Symbolic dynamics

The Symbolic dynamics is a branch of the theory of dynamical systems, in the methods of formal languages ​​(grammar theory, automata theory, complexity theory ) and the theory of stochastic processes are used.

The starting point of the symbolic dynamics is a discrete-time dynamic system with state space and flow, with either the same or for reversible dynamics is the same. By a partition of the state space into a finite number of n subsets one gains a rule, such an initial condition is to be mapped to a symbol sequence:

, The initial condition to a symbol, if so, as the successor state to a symbol, if, in short, the way state by a symbol to when. The result of the trajectory traversed by the subsets can then be regarded as a symbol sequence with symbols. Here, a finite alphabet is composed of as many symbols as there are subsets of the partition.

Depending on the amount of time to obtain either side endless symbol sequences, if (English shifts one-sided ) or double-sided symbol sequences infinite if (English shifts two-sided ). The point after usually indicates the initial condition. The set of symbol sequences, the state space of symbolic dynamics is then ( one-sided), or written. The above design procedure of a sequence of symbols corresponds to a picture, so that when, with the subset of the partition is associated with the icon.

A simple connection between the symbolic representations of the initial condition and its first iteration: As is represented by the sequence, the design of the sequence of symbols for the symbol starts. Therefore, it is represented by the sequence. So different from the fact that all symbols are moved into one place to the left ( or the point one place to the right). Therefore, there is an image on the space of the symbol sequences with. The figure is called a left shift (English left- shift). hot symbolic dynamics. Is the connection between the original system and the symbolic dynamics.

Credentials

  • T. Schürmann and I. Hoffmann, The entropy of ' strange ' billiards inside n -simplex, J. Phys A: Math Gen. 28 (1995) 5033-5039.
  • T. Schürmann, Scaling behavior of entropy estimates, J. Phys A: Math Gen. 35 (2002) 1589-1596.
  • Gerald Teschl: Ordinary Differential Equations and Dynamical Systems ( Graduate Studies in Mathematics = 140. ). American Mathematical Society, Providence, 2012 ISBN 978-0-8218-8328-0 ( free online version).
  • Ordinary Differential Equations
  • Stochastics
Formal grammar Reversibility
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