15,474 research outputs found
Denjoy constructions for fibred homeomorphisms of the torus
We construct different types of quasiperiodically forced circle
homeomorphisms with transitive but non-minimal dynamics. Concerning the recent
Poincar\'e-like classification for this class of maps of Jaeger-Stark, we
demonstrate that transitive but non-minimal behaviour can occur in each of the
different cases. This closes one of the last gaps in the topological
classification. Actually, we are able to get some transitive quasiperiodically
forced circle homeomorphisms with rather complicated minimal sets. For example,
we show that, in some of the examples we construct, the unique minimal set is a
Cantor set and its intersection with each vertical fibre is uncountable and
nowhere dense (but may contain isolated points). We also prove that minimal
sets of the later kind cannot occur when the dynamics are given by the
projective action of a quasiperiodic SL(2,R)-cocycle. More precisely, we show
that, for a quasiperiodic SL(2,R)-cocycle, any minimal strict subset of the
torus either is a union of finitely many continuous curves, or contains at most
two points on generic fibres
Combinatorial Properties of Triangle-Free Rectangle Arrangements and the Squarability Problem
We consider arrangements of axis-aligned rectangles in the plane. A geometric
arrangement specifies the coordinates of all rectangles, while a combinatorial
arrangement specifies only the respective intersection type in which each pair
of rectangles intersects. First, we investigate combinatorial contact
arrangements, i.e., arrangements of interior-disjoint rectangles, with a
triangle-free intersection graph. We show that such rectangle arrangements are
in bijection with the 4-orientations of an underlying planar multigraph and
prove that there is a corresponding geometric rectangle contact arrangement.
Moreover, we prove that every triangle-free planar graph is the contact graph
of such an arrangement. Secondly, we introduce the question whether a given
rectangle arrangement has a combinatorially equivalent square arrangement. In
addition to some necessary conditions and counterexamples, we show that
rectangle arrangements pierced by a horizontal line are squarable under certain
sufficient conditions.Comment: 15 pages, 13 figures, extended version of a paper to appear at the
International Symposium on Graph Drawing and Network Visualization (GD) 201
Reidemeister/Roseman-type Moves to Embedded Foams in 4-dimensional Space
The dual to a tetrahedron consists of a single vertex at which four edges and
six faces are incident. Along each edge, three faces converge. A 2-foam is a
compact topological space such that each point has a neighborhood homeomorphic
to a neighborhood of that complex. Knotted foams in 4-dimensional space are to
knotted surfaces, as knotted trivalent graphs are to classical knots. The
diagram of a knotted foam consists of a generic projection into 4-space with
crossing information indicated via a broken surface. In this paper, a finite
set of moves to foams are presented that are analogous to the Reidemeister-type
moves for knotted graphs. These moves include the Roseman moves for knotted
surfaces. Given a pair of diagrams of isotopic knotted foams there is a finite
sequence of moves taken from this set that, when applied to one diagram
sequentially, produces the other diagram.Comment: 18 pages, 29 figures, Be aware: the figure on page 3 takes some time
to load. A higher resolution version is found at
http://www.southalabama.edu/mathstat/personal_pages/carter/Moves2Foams.pdf .
If you want to use to any drawings, please contact m
Extremal Values of the Interval Number of a Graph
The interval number of a simple graph is the smallest number such that to each vertex in there can be assigned a collection of at most finite closed intervals on the real line so that there is an edge between vertices and in if and only if some interval for intersects some interval for . The well known interval graphs are precisely those graphs with . We prove here that for any graph with maximum degree . This bound is attained by every regular graph of degree with no triangles, so is best possible. The degree bound is applied to show that for graphs on vertices and for graphs with edges
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