We study definable sets D of SU-rank 1 in Meq, where M is a countable homogeneous and simple structure in a language with finite relational vocabulary. Each such D can be seen as a 'canonically embedded structure', which inherits all relations on D which are definable in Meq, and has no other definable relations. Our results imply that if no relation symbol of the language of M has arity higher than 2, then there is a close relationship between triviality of dependence and D being a reduct of a binary random structure. Somewhat more precisely: (a) if for every n≥2, every n-type p(x1,...,xn) which is realized in D is determined by its sub-2-types q(xi,xj)⊆p, then the algebraic closure restricted to D is trivial; (b) if M has trivial dependence, then D is a reduct of a binary random structure.

We investigate the symbolic dynamics for the double standard maps of the circle onto itself, given by f(a,b) (x) = 2x + a + (b/pi) sin(2 pi x) (mod 1), where b = 1 and a is a real parameter, 0 <= a < 1

We study a certain type of action of categories on categories and on operads. Using the structure of the categories Δ and Ω governing category and operad structures, respectively, we define categories which instead encode the structure of a category acting on a category, or a category acting on an operad. We prove that the former has the structure of an elegant Reedy category, whereas the latter has the structure of a generalized Reedy category. In particular, this approach gives a new way to regard group actions on categories and on operads.

We discuss the exactness of estimates in the finite sum theorems for transfinite inductive dimensions trind and trInd. The technique obtained gives an opportunity to repeat and sometimes strengthen some well known results about compacta with trind not equal trInd. In particular we improve an estimate of the small transfinite inductive dimension of Smirnov's compacts S-alpha, alpha < omega(1), given by Luxemburg [Lu2].

5.

Chatyrko, VA

et al.

Linköping University, Department of Mathematics. Linköping University, The Institute of Technology.

Hattori, Y

Dept Math, Matsue, Shimane 6908504, Japan.

On a question of de Groot and Nishiura2002In: Fundamenta Mathematicae, ISSN 0016-2736, E-ISSN 1730-6329, Vol. 172, no 2, p. 107-115Article in journal (Refereed)

Abstract [en]

Let Z(n) = [0, 1](n+1) \ (0, 1)(n) x {0}. Then cmp Z(n) < def Z(n) for n greater than or equal to 5. This is the answer to a question posed by de Groot and Nishiura [GN] for n greater than or equal to 5.

6.

Laksov, Dan

KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).

Various kinds of radicals of ideals in commutative rings with identity appear in many parts of algebra and geometry, in particular in connection with the Hilbert Nullstellensatz, both in the noetherian and the non-noetherian case. All of these radicals, except the *-radicals, have the fundamental, and very useful, property that the radical of an ideal is the intersection of radical primes, that is, primes that are equal to their own radical. It is easy to verify that when the ring A is noetherian then the *-radical R(J) of an ideal is the intersection of *-radical primes. However, it has been an open question whether this holds in general. The main purpose of this article is to give an example of a ring with a *-radical that is not radical. To our knowledge it is the first example of a natural radical on a ring such that the radical of each ideal is not the intersection of radical primes. More generally, we present a method that may be used to construct more such examples. The main new idea is to introduce radical operations on the closed sets of topological spaces. We can then use the Zariski topology on the spectrum of a ring to translate algebraic questions into topology. It turns out that the quite intricate algebraic manipulations involved in handling the *-radical become much more transparent when rephrased in geometric terms.

7.

Miles, Richard

KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).

This paper is concerned with the entropy of an action of a countable torsion-free abelian group G by continuous automorphisms of a compact abelian group X. A formula is obtained that expresses the entropy in terms of the Mahler measure of a greatest common divisor, complementing earlier work by Einsiedler, Lind, Schmidt and Ward. This leads to a uniform method for calculating entropy whenever G is free. In cases where these methods do not apply, a possible entropy formula is conjectured. The entropy of subactions is examined and, using a theorem of P. Samuel, it is shown that a mixing action of an infinitely generated group of finite rational rank cannot have a finitely generated subaction with finite non-zero entropy. Applications to the concept of entropy rank are also considered.

8. Misiurewicz, Michal

et al.

Rodrigues, Ana

KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).

Making use of the Nielsen fixed point theory, we study a conjugacy invariant of braids, which we call the level index function. We present a simple algorithm for computing it for positive permutation cyclic braids.

Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics.

Cherry flows with non-trivial attractors2019In: Fundamenta Mathematicae, ISSN 0016-2736, E-ISSN 1730-6329, Vol. 244, no 3, p. 243-253Article in journal (Refereed)

Abstract [en]

We provide an example of a Cherry flow (i.e. a C-infinity flow on the 2-dimensional torus with a sink and a saddle) having a quasi-minimal set which is an attractor. The first return map for such a flow, also constructed in the paper, is a C-infinity circle map having a flat interval and a non-trivial wandering interval.