32,614 research outputs found
A weak equivalence between shifts of finite type
International audienc
Strong shift equivalence, algebraic K-theory, and isolating zero-dimensional dynamics on manifolds
We study the relations of shift equivalence and strong shift equivalence for matrices over a ring , and establish a connection between these relations and algebraic K-theory. We utilize this connection to obtain results in two areas where the shift and strong shift equivalence relations play an important role: the study of finite group extensions of shifts of finite type, and the Generalized Spectral Conjectures of Boyle and Handelman for nonnegative matrices over subrings of the real numbers.
We show the refinement of the shift equivalence class of a matrix over a ring by strong shift equivalence classes over the ring is classified by a quotient of the algebraic K-group NK_{1}(\calR). We use the K-theory of non-commutative localizations to show that in certain cases the subgroup must vanish, including the case is invertible over .
We use the K-theory connection to clarify the structure of algebraic invariants for finite group extensions of shifts of finite type. In particular, we give a strong negative answer to a question of Parry, who asked whether the dynamical zeta function determines up to finitely many topological conjugacy classes the extensions by of a fixed mixing shift of finite type.
We apply the K-theory connection to prove the equivalence of a strong and weak form of the Generalized Spectral Conjecture of Boyle and Handelman for primitive matrices over subrings of .
We construct explicit matrices whose class in the algebraic K-group is non-zero for certain rings motivated by applications.
We study the possible dynamics of the restriction of a homeomorphism of a compact manifold to an isolated zero-dimensional set. We prove that for every compact zero-dimensional system can arise as an isolated invariant set for a homeomorphism of a compact -manifold. In dimension two, we provide obstructions and examples
The mathematical research of William Parry FRS
In this article we survey the mathematical research of the late William (Bill) Parry, FRS
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
The ghost length and duality on the chain and cochain type levels
We establish equalities between cochain and chain type levels of maps by
making use of exact functors which connect appropriate derived and coderived
categories. Relevant conditions for levels of maps to be finite are extracted
from the equalities which we call duality on the levels. Moreover, we give a
lower bound of the cochain type level of the diagonal map on the classifying
space of a Lie group by considering the ghostness of a shriek map which appears
in derived string topology. A variant of Koszul duality for a differential
graded algebra is also discussed.Comment: 23 pages. This is a new verision of the preprint "Duality on the
(co)chain type levels". The title is change
On differential graded categories
Differential graded categories enhance our understanding of triangulated
categories appearing in algebra and geometry. In this survey, we review their
foundations and report on recent work by Drinfeld, Dugger-Shipley, ..., Toen
and Toen-Vaquie.Comment: 30 pages, correction at the end of 3.9, corrections and added
references in 5.
G-symmetric spectra, semistability and the multiplicative norm
In this paper we develop the basic homotopy theory of G-symmetric spectra
(that is, symmetric spectra with a G-action) for a finite group G, as a model
for equivariant stable homotopy with respect to a G-set universe. This model
lies in between Mandell's equivariant symmetric spectra and the G-orthogonal
spectra of Mandell and May and is Quillen equivalent to the two. We further
discuss equivariant semistability, construct model structures on module,
algebra and commutative algebra categories and describe the homotopical
properties of the multiplicative norm in this context.Comment: Final published versio
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