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Continuous Itinerary Functions in Higher Dimensions
by
Stewart Baldwin
Auburn University, Auburn, AL
Given f:X ® X and a partition {Sa: a Î S} of X (where S is a set of symbols), the itinerary of a point x Î X with respect to the function f and the partition is defined to be the unique sequence a Î Sw such that fn(x) Î San for all n Î w = {0, 1, 2, ...}. The natural map q:X ® S defined by x Î Sq(x) induces a natural quotient topology on S, called the symbol topology, and the corresponding product topology on Sw is called the itinerary topology. A natural consequence of this definition is that in this topology the itinerary function is a continuous function from X into Sw.
In previous talks, it has been shown how these itinerary topologies can be a useful tool in one-dimensional dynamical systems, despite the fact that these topologies are non-Hausdorff in most cases. We discuss how these results can be extended to higher dimensions. For example, if we define a turning point of f:X ® X to be any point t such that every neighborhood of t contains an arc on which f is not one-to-one, and let T be the set of turning points of f, then it will often be the case that the dynamics of f on X can be recovered from the topology on S and the set of itineraries of points in T.
Date received: February 28, 2005
Copyright © 2005 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # capc-72.