Category Theory of Symbolic Dynamics

We study the central objects of symbolic dynamics, that is, subshifts and block maps, from the perspective of basic category theory, and present several natural categories with subshifts as objects and block maps as morphisms. Our main goals are to find universal objects in these symbolic categories, to classify their block maps based on their category theoretic properties, to prove category theoretic characterizations for notions arising from symbolic dynamics, and to establish as many natural properties (finite completeness, regularity etc.) as possible. Existing definitions in category theory suggest interesting new problems in symbolic dynamics. Our main technical contributions are the solution to the dual problem of the Extension Lemma and results on certain types of conserved quantities, suggested by the concept of a coequalizer.</p

Predecessor-successor transitions in institutional and interpersonal contexts: on the development of a theory of transfer of personal objects

"This article outlines the development of a theory of predecessor successor transitions in social contexts using a grounded theory approach. The theory can be applied to such diverse phenomena as the transfer of family businesses to the next generation, university chair succession, the passing on of parental roles (for example in the case of adoption or remarriage), and organ transplantation. The core conceptual category that emerged was 'the transfer of personal objects'. This concept refers to the transfer of the power of disposal over objects that are fundamental to the identity and the identification of the owner. A number of theoretical dimensions of the category were identified. Methodologically speaking, the theory generated can be classified as a formal grounded theory. In other words, the comparison of different empirical fields and cases using hermeneutical analysis yielded a transdisciplinary social science category that can be employed to conceptualize the dynamics of the development of interpersonal, social, or institutional structures, especially with regard to the links and the interplay between material and symbolic components, between the individual and the social, and the past and the present." (author's abstract

Genericity in Topological Dynamics

We study genericity of dynamical properties in the space of homeomorphisms of the Cantor set and in the space of subshifts of a suitably large shift space. These rather different settings are related by a Glasner-King type correspondence: genericity in one is equivalent to genericity in the other. By applying symbolic techniques in the shift-space model we derive new results about genericity of dynamical properties for transitive and totally transitive homeomorphisms of the Cantor set. We show that the isomorphism class of the universal odometer is generic in the space of transitive systems. On the other hand, the space of totally transitive systems displays much more varied dynamics. In particular, we show that in this space the isomorphism class of every Cantor system without periodic points is dense, and the following properties are generic: minimality, zero entropy, disjointness from a fixed totally transitive system, weak mixing, strong mixing, and minimal self joinings. The last two stand in striking contrast to the situation in the measure-preserving category. We also prove a correspondence between genericity of dynamical properties in the measure-preserving category and genericity of systems supporting an invariant measure with the same property.Comment: 48 pages, to appear in Ergodic Theory Dynamical Systems. v2: revised exposition, added proof that the universal odometer is generic among transitive Cantor homeomorphism

Strong shift equivalence as a category notion

In this paper, we present a completely radical way to investigate the main problem of symbolic dynamics, the conjugacy problem, by proving that this problem actually relates to a natural question in category theory regarding the theory of traced bialgebras. As a consequence of this theory, we obtain a systematic way of obtaining new invariants for the conjugacy problem by looking at existing bialgebras in the literature

Strong shift equivalence as a category notion

In this paper, we present a completely radical way to investigate the main problem of symbolic dynamics, the conjugacy problem, by proving that this problem actually relates to a natural question in category theory regarding the theory of traced bialgebras. As a consequence of this theory, we obtain a systematic way of obtaining new invariants for the conjugacy problem by looking at existing bialgebras in the literature

Topological Quantum Field Theory And Strong Shift Equivalence

Given a TQFT in dimension d+1, and an infinite cyclic covering of a closed (d+1)-dimensional manifold M, we define an invariant taking values in a strong shift equivalence class of matrices. The notion of strong shift equivalence originated in R. Williams' work in symbolic dynamics. The Turaev-Viro module associated to a TQFT and an infinite cyclic covering is then given by the Jordan form of this matrix away from zero. This invariant is also defined if the boundary of M has a circle factor and the infinite cyclic cover of the boundary is standard. We define a variant of a TQFT associated to a finite group G which has been studied by Quinn. In this way, we recover a link invariant due to D. Silver and S. Williams. We also obtain a variation on the Silver-Williams invariant, by using the TQFT associated to G in its unmodified form.Comment: AMS-TeX, 8 pages, a few small changes change