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The symbolic dynamics technique is well-known for low-dimensional dynamicalsystems and chaotic maps, and lies at the roots of the thermodynamic formalismof dynamical systems. Here we show that this technique can also be successfullyapplied to time series generated by complex systems of much higherdimensionality. Our main example is the investigation of share price returns ina coarse-grained way. A nontrivial spectrum of Renyi entropies is found. Westudy how the spectrum depends on the time scale of returns, the sector ofstocks considered, as well as the number of symbols used for the symbolicdescription. Overall our analysis confirms that in the symbol space transitionprobabilities of observed share price returns depend on the entire history ofprevious symbols, thus emphasizing the need for a modelling based onnon-Markovian stochastic processes. Our method allows for quantitativecomparisons of entirely different complex systems, for example the statisticsof symbol sequences generated by share price returns using 4 symbols can becompared with that of genomic sequences.

Symbolic dynamics techniques for complex systems: Application to share price dynamics