SYMBOLIC DYNAMICS AND CHARACTERIZATION OF COMPLEXITY

Symbolic dynamics deals with robust properties of dynamics without digging into numbers. For one-dimensional maps there have been some new results (periodic window theorem, generalized composition rule, construction of median words without using harmonics and antiharmonics, incorporation of discrete symmetry to analyze symmetry breaking and symmetry restoration, etc.) This new understanding has been applied to ordinary differential equations, in particular, the systematics of stable periodic solutions in the Lorenz model has been shown to be given basically by symbolic dynamics of the cubic map. Symbolic dynamics may be used to extract invariant characteristics from time series. Its relation with grammatical complexity will also be commented on.