On the Topological Entropy of Multivalued Maps

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CSSS 2006 Santa Fe

The notion of topological entropy, an invariant quantitative measure of the complexity of a dynamical system f, is often used to define chaos. It measures the extent to which points that are very near are mapped to points that are far away by repeated application of f. A dynamical system f may be said to be chaotic if its topological entropy, denoted by h(f), is positive. Although topological entropy has already been established for more general single-valued maps, in general, many chaotic systems encountered in practice cannot be described by such maps. We consider the topological entropy of maps that cannot be described by one-dimensional dynamics.

Given arbitrary sets X and Y, a multivalued map F from X to Y is such that F(x) is assigned a subset Y' of Y for all x in X. A single-valued map g from X'(subset of X) to Y is called a selector for F over X' if g(x) is in F(x) for all x in X'. We have two objectives here, namely (1) to give a precise definition of the topological entropy of a multivalued map and (2) to show that picking a continuous selector for F that resembles a one-dimensional map is sufficient to give a lower bound for the topological entropy of F. More precisely, we consider F to be reconstruction map (of some unknown dynamical system f) in two-dimensional space. We employ the method of time-delay reconstruction and verify (2) using experimental data that, when reconstructed in two-dimensional space, can be viewed as a union of continuous selectors.

Vivien, (insert name if you have interesting data),