121 research outputs found
Sofic-Dyck shifts
We define the class of sofic-Dyck shifts which extends the class of
Markov-Dyck shifts introduced by Inoue, Krieger and Matsumoto. Sofic-Dyck
shifts are shifts of sequences whose finite factors form unambiguous
context-free languages. We show that they correspond exactly to the class of
shifts of sequences whose sets of factors are visibly pushdown languages. We
give an expression of the zeta function of a sofic-Dyck shift
Zeta functions and topological entropy of the Markov-Dyck shifts
The Markov-Dyck shifts arise from finite directed graphs. An expression for the zeta function of a Markov-Dyck shift is given. The derivation of this expression is based on a formula in Keller [12]. For a class of examples that includes the Fibonacci-Dyck shift the zeta functions and topological entropy are determined
Direct topological factorization for topological flows
This paper considers the general question of when a topological action of a
countable group can be factored into a direct product of a nontrivial actions.
In the early 1980's D. Lind considered such questions for -shifts
of finite type. We study in particular direct factorizations of subshifts of
finite type over and other groups, and -subshifts
which are not of finite type. The main results concern direct factors of the
multidimensional full -shift, the multidimensional -colored chessboard
and the Dyck shift over a prime alphabet.
A direct factorization of an expansive -action must be finite,
but a example is provided of a non-expansive -action for which
there is no finite direct prime factorization. The question about existence of
direct prime factorization of expansive actions remains open, even for
.Comment: 21 pages, some changes and remarks added in response to suggestions
by the referee. To appear in ETD
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Affine permutations and rational slope parking functions
We introduce a new approach to the enumeration of rational slope parking functions with respect to the area and a generalized dinv statistics, and relate the combinatorics of parking functions to that of affine permutations. We relate our construction to two previously known combinatorial constructions: Haglund’s bijection ζ exchanging the pairs of statistics (area, dinv) and (bounce, area) on Dyck paths, and the Pak-Stanley labeling of the regions of k-Shi hyperplane arrangements by k-parking functions. Essentially, our approach can be viewed as a generalization and a unification of these two constructions. We also relate our combinatorial constructions to representation theory. We derive new formulas for the Poincaré polynomials of certain affine Springer fibers and describe a connection to the theory of finite-dimensional representations of DAHA and non-symmetric Macdonald polynomials
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