Zbigniew Nitecki (Tufts University)

Preimage entropy and symbolic dynamics.

Abstract. This is a joint work with Doris Fiebig and Ulf-Rainer Fiebig. Topological entropy h_top is based on the dispersion of orbits in forward time. For noninvertible maps, several conjugacy invariants based on preimage structure have been formulated and studied. Pointwise preimage entropy h_m is the growth rate of the maximum number of $(n,\epsilon)$-separated n^th preimages of a point in the space, while branch preimage entropy h_b is the growth rate of the number of $(n,\epsilon)$-distinct preimage sets of points.

Hurley showed that $h_m\leq h_{top}\leq h_m + h_b$, and examples show that either inequality can be strict. For a (one-sided) subshift, h_m is the growth rate of the size of the largest "n^th predecessor set" while h_b is the growth rate of the number of distinct such sets.

Theorem 1: For a subshift (and more generally for a forward-expansive map on a compact metric space), h_m=h_top and there exists an entropy point: a point for which the number of n^th preimages grows at precisely this rate. A more general notion of "entropy point" can be formulated, and such points can be shown to exist for any asymptotically h-expansive map. However, we construct an "almost" h-expansive map with no entropy point--in fact, for this map the number of $(n,\epsilon)$-separated n^th preimages of any particular point grows subexponentially, but h_m>0. (This answers a question raised by Hurley.)

A second result concerns inverse limits (or natural extensions). It is well known that non-conjugate systems (even shifts) can have conjugate inverse limits.

Theorem 2: Forward-expansive surjections on compact metric spaces (in particular subshifts) whose inverse limits are conjugate have equal branch preimage entropy.