Symbolic dynamism

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

The starting point of the symbolic dynamics is a time-discrete dynamic system with state space and flow , where either is equal or for reversible dynamics is equal . By partitioning the state space into a finite number of n subsets , one obtains a rule how an initial condition is to be mapped onto a symbol sequence : ${\displaystyle (X,\Phi )}$ ${\displaystyle X}$ ${\displaystyle \Phi \colon \mathbb {T} \times X\to X}$ ${\displaystyle \mathbb {T} }$ ${\displaystyle \mathbb {N} }$ ${\displaystyle \mathbb {Z} }$ ${\displaystyle X}$ ${\displaystyle A_{1},A_{2},\dots A_{n}}$ ${\displaystyle x_{0}\in X}$

As the initial condition a symbol to when , as then the next state a symbol to when , in short, as the state symbol to when . The sequence of subsets traversed by the trajectory curve can then be viewed as a symbol sequence with symbols . A finite alphabet consists of as many symbols as there are subsets of the partition. ${\displaystyle x_{0}}$ ${\displaystyle a_{k_{0}}}$ ${\displaystyle x_{0}\in A_{k_{0}}}$ ${\displaystyle x_{1}=\Phi (1,x_{0})}$ ${\displaystyle a_{k_{1}}}$ ${\displaystyle x_{1}\in A_{k_{1}}}$ ${\displaystyle x_{t}=\Phi (t,x_{0})}$ ${\displaystyle a_{k_{t}}}$ ${\displaystyle x_{t}\in A_{k_{t}}}$ ${\displaystyle \{x_{t}|t\in \mathbb {T} \}}$ ${\displaystyle s=a_{k_{0}}.a_{k_{1}}a_{k_{2}}a_{k_{3}}\dots }$ ${\displaystyle a_{k_{t}}\in \mathbf {A} }$ ${\displaystyle \mathbf {A} }$

Depending on the amount of time , one obtains either one-sided infinite symbol sequences if ( one-sided shifts ), or two-sided infinite symbol sequences if ( two-sided shifts ). The point after usually indicates the initial condition. The set of symbol sequences, the state space of the symbolic dynamics, is then (one-sided) or written. The above construction rule for a symbol sequence then corresponds to a mapping , so that , if , the subset of the partition is assigned the symbol . ${\displaystyle \mathbb {T} }$ ${\displaystyle s=s_{0}.s_{1}s_{2}\dots }$ ${\displaystyle \mathbb {T} =\mathbb {N} }$ ${\displaystyle s=\dots s_{-2}s_{-1}s_{0}.s_{1}s_{2}\dots }$ ${\displaystyle \mathbb {T} =\mathbb {Z} }$ ${\displaystyle s_{0}}$ ${\displaystyle \Sigma =\mathbf {A} ^{\mathbb {N} }}$ ${\displaystyle \Sigma =\mathbf {A} ^{\mathbb {Z} }}$ ${\displaystyle \pi \colon X\to \Sigma }$ ${\displaystyle \pi (x_{0})=s}$ ${\displaystyle \Phi (t,x_{0})\in A_{k_{t}}}$ ${\displaystyle A_{k_{t}}}$ ${\displaystyle a_{k_{t}}\in \mathbf {A} }$

There is a simple connection between the symbolic representations of an initial condition and its first iteration : While the sequence is used to represent, the construction of the symbol sequence begins with the symbol . Hence it is represented by the sequence . differs from in that all symbols have been moved one place to the left (or the point has been moved one place to the right). Therefore there is an illustration in the space of the symbol sequences , with . The figure is left shift (Engl. Left-shift called). are called symbolic dynamics. Between the original system and the symbolic dynamics is the connection . ${\displaystyle x_{0}}$ ${\displaystyle x_{1}=\Phi (1,x_{0})}$ ${\displaystyle x_{0}}$ ${\displaystyle s=s_{0}.s_{1}s_{2}s_{3}\dots }$ ${\displaystyle x_{1}}$ ${\displaystyle s_{1}}$ ${\displaystyle x_{1}}$ ${\displaystyle s'=s_{1}.s_{2}s_{3}s_{4}\dots }$ ${\displaystyle s'}$ ${\displaystyle s}$ ${\displaystyle s}$ ${\displaystyle \sigma \colon \Sigma \to \Sigma }$ ${\displaystyle \sigma (s)=s'}$ ${\displaystyle \sigma }$ ${\displaystyle (\Sigma ,\sigma )}$ ${\displaystyle (X,\Phi )}$ ${\displaystyle (\Sigma ,\sigma )}$ ${\displaystyle \pi \circ \Phi =\sigma \circ \pi }$

literature

• T. Schürmann, I. Hoffmann: arxiv : nlin / 0208048 In: J. Phys A: Math. Gen. , 28, 1995, pp. 5033-5039.
• T. Schürmann: Scaling behavior of entropy estimates . In: J. Phys A: Math. Gen. , 35, 2002, pp. 1589-1596, arxiv : cond-mat / 0203409v1
• Gerald Teschl : Ordinary Differential Equations and Dynamical Systems (=  Graduate Studies in Mathematics . Volume 140 ). American Mathematical Society, Providence 2012, ISBN 978-0-8218-8328-0 ( mat.univie.ac.at ).