What symbolic dynamics do we get with a misplaced partition? On the validity of threshold crossings analysis of chaotic time-series

An increasingly popular method of encoding chaotic time-series from physica l experiments is the so-called threshold crossings technique, where one sim ply replaces the real valued data with symbolic data of relative positions to an arbitrary partition at discrete times. The implication has been that this symbolic encoding describes the original dynamical system. On the othe r hand, the literature on generating partitions of non-hyperbolic dynamical systems has shown that a good partition is non-trivial to find. It is beli eved that the generating partition of non-uniformly hyperbolic dynamical sy stem connects "primary tangencies", which are generally not simple Lines as used by a threshold crossings. Therefore, we investigate consequences of u sing itineraries generated by a non-generating partition. We do most of our rigorous analysis using the tent map as a benchmark example, but show nume rically that our results likely generalize. In summary, we find the misrepr esentation of the dynamical system by "sample-path" symbolic dynamics of an arbitrary partition can be severe, including (sometimes extremely) diminis hed topological entropy, and a high degree of non-uniqueness. Interestingly , we find topological entropy as a function of misplacement to be devil's s taircase-like, but surprisingly non-monotone. (C) 2001 Elsevier Science B.V . All right reserved.