# Blog Archives

# Topic Archive: dynamical systems

# Effective dimension in subshifts

A subshift is a subset of Cantor space that is both topologically closed and closed under the shift operation. Any element of a subshift can be viewed as a trajectory in a discrete dynamical system. One way a trajectory can be complicated is to have a large effective dimension. We consider the effective dimension spectrum of a subshift *X*, defined as { dim *x : x ∈ X* }, where dim is the effective dimension. Most commonly, the dimension spectrum of *X* is [ *0, h(X)* ], where *h(X)* is the entropy of *X*. We give examples where the dimension spectrum does not follow this pattern, and discuss partial results and open questions related to the problem of characterizing the dimension spectra of subshifts. No prior knowledge about subshifts will be assumed.