![]() For example, the motivation for the original rank-one transformation constructed by Chacon Reference 5 was to build a measure preserving transformation that is weakly mixing but not mixing. Because the concept of rank-one subshifts came from rank-one transformations, the study of their topological dynamical properties is often motivated by their ergodic-theoretic counterparts which tend to have a long history. In Reference 12 the current authors studied the topological mixing properties of rank-one subshifts. This led to the definition of rank-one subshifts, which was first studied by the first author and Hill in Reference 11, where they gave a characterization for the topological isomorphism relation of rank-one subshifts based on the cutting and spacer parameters. However, the constructive symbolic definition seemed to behave somewhat differently from the other definitions, which led to further research from the perspective of symbolic and topological dynamics, such as in Reference 2, Reference 8, and Reference 9.īecause the constructive symbolic definition works with a shift space, it was natural to study systems coming from the constructive symbolic definition in the setting of topological dynamics. Many rank-one transformations could be shown to satisfy each of the different definitions. Ferenczi Reference 10 was a comprehensive survey summarizing many results and systematically studying several different definitions of rank-one transformations that had appeared in the literature. Since then, rank-one transformations have come up often as important examples and counterexamples in ergodic theory and have been studied extensively by many researchers. In 1965, Chacon Reference 5 introduced the concept of rank-one measure-preserving transformations and constructed the first examples. Thus, we completely characterize the subshift factors of rank-one subshifts. We study topological factors of rank-one subshifts and prove that those factors that are themselves subshifts are either finite or isomorphic to the original rank-one subshifts. Topological factors of rank-one subshifts
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