Racks and blocked braids

In the paper Blocked-braid Groups, submitted to Applied Categorical Structures, the present authors together with Davide Maglia introduced the blocked-braid groups BB_n on n strands, and proved that a blocked torsion has order either 2 or 4. We conjectured that the order was actually 4 but our methods in that paper, which involved introducing for any group G a braided monoidal category of tangled relations, were inadequate to demonstrate this fact. Subsequently Davide Maglia in unpublished work investigated exactly what part of the structure and properties of a group G are needed to permit the construction of a braided monoidal category with a tangle algebra and was able to distinguish blocked two-torsions from the identity. In this paper we present a simplification of his answer, which turns out to be related to the notion of rack. We show that if G is a rack then there is a braided monoidal category TRel_G generalizing that of the above paper. Further we introduce a variation of the notion of rack which we call irack which yields a tangle algebra in TRel_G. Iracks are in particular racks but have in addition to the operations abstracting group conjugation also a unary operation abstracting group inverse. Using iracks we obtain new invariants for tangles and blocked braids permitting us to present a proof of Maglia's result that a blocked double torsion is not the identity. This work was presented at the Conference in Memory of Aurelio Carboni, Milan, 24-26 June 2013

Tangled Circuits

The theme of the paper is the use of commutative Frobenius algebras in braided strict monoidal categories in the study of varieties of circuits and communicating systems which occur in Computer Science, including circuits in which the wires are tangled. We indicate also some possible novel geometric interest in such algebras

Blocked-braid Groups

We introduce and study a family of groups $\mathbf{BB}_n$, called the blocked-braid groups, which are quotients of Artin's braid groups $\mathbf{B}_n$, and have the corresponding symmetric groups $\Sigma_n$ as quotients. They are defined by adding a certain class of geometrical modifications to braids. They arise in the study of commutative Frobenius algebras and tangle algebras in braided strict monoidal categories. A fundamental equation true in $\mathbf{BB}_n$ is Dirac's Belt Trick; that torsion through $4\pi$ is equal to the identity. We show that $\mathbf{BB}_n$ is finite for $n=1,2$ and 3 but infinite for $n>3$

The compositional construction of Markov processes II

In an earlier paper we introduced a notion of Markov automaton, together with parallel operations which permit the compositional description of Markov processes. We illustrated by showing how to describe a system of n dining philosophers, and we observed that Perron-Frobenius theory yields a proof that the probability of reaching deadlock tends to one as the number of steps goes to infinity. In this paper we add sequential operations to the algebra (and the necessary structure to support them). The extra operations permit the description of hierarchical systems, and ones with evolving geometry

Behavior of the Escape Rate Function in Hyperbolic Dynamical Systems

For a fixed initial reference measure, we study the dependence of the escape rate on the hole for a smooth or piecewise smooth hyperbolic map. First, we prove the existence and Holder continuity of the escape rate for systems with small holes admitting Young towers. Then we consider general holes for Anosov diffeomorphisms, without size or Markovian restrictions. We prove bounds on the upper and lower escape rates using the notion of pressure on the survivor set and show that a variational principle holds under generic conditions. However, we also show that the escape rate function forms a devil's staircase with jumps along sequences of regular holes and present examples to elucidate some of the difficulties involved in formulating a general theory.Comment: 21 pages. v2 differs from v1 only by additions to the acknowledgment

Invariant Measures and Decay of Correlations for a Class of Ergodic Probabilistic Cellular Automata

We give new sufficient ergodicity conditions for two-state probabilistic cellular automata (PCA) of any dimension and any radius. The proof of this result is based on an extended version of the duality concept. Under these assumptions, in the one dimensional case, we study some properties of the unique invariant measure and show that it is shift-mixing. Also, the decay of correlation is studied in detail. In this sense, the extended concept of duality gives exponential decay of correlation and allows to compute explicitily all the constants involved