1,983 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
Topological conjugation classes of tightly transitive subgroups of
Let denote the group of orientation
preserving homeomorphisms of the circle . A subgroup of
is tightly transitive if it is topologically
transitive and no subgroup of with has this property;
is almost minimal if it has at most countably many nontransitive points. In the
paper, we determine all the topological conjugation classes of tightly
transitive and almost minimal subgroups of
which are isomorphic to for any integer .Comment: 17 pages, 4 figure
Homogeneous Transformation Groups of the Sphere
In this paper, we study the structure of homogeneous subgroups of the
homeomorphism group of the sphere, which are defined as closed groups of
homeomorphisms of the sphere that contain the rotation group. We prove two
structure theorems about the behaviour and properties of such groups and
present a diagram of the structure of these groups partly on the basis of these
results. In addition, we prove a number of explicit relations between the
groups in the diagram.Comment: This paper has been withdrawn by an irreconcilable difference of
opinion between the authors on its presentation and content
Surface homeomorphisms : the interplay between topology, geometry and dynamics
In this thesis we study certain classes of surface homeomorphisms and in particular the interplay between the topology of the underlying surface and topological, geometrical and dynamical properties of the homeomorphisms. We study three problems in three independent chapters: The first problem is to describe the minimal sets of non-resonant torus homeomorphisms, i.e. those homeomorphisms which are in a sense close to a minimal translation of the torus. We study the possible minimal sets that such a homeomorphism can admit, uniqueness of minimal sets and their relation with other limit sets. Further, we give examples of homeomorphisms to illustrate the possible dynamics. In a sense, this study is a two-dimensional analogue of H. Poincar´e’s study of orbit structures of orientation preserving circle homeomorphisms without periodic points.
The second problem concerns the interplay between smoothness of surface diffeomorphisms, entropy and the existence of wandering domains. Every surface admits homeomorphisms with positive entropy that permutes a dense collection of domains that have bounded geometry. However, we show that at a certain level of differentiability it becomes impossible for a diffeomorphism of a surface to have positive entropy and permute a dense collection of domains that has bounded geometry.
The third problem concerns quasiconformal homogeneity of surfaces; i.e., whether a surface admits a transitive family of quasiconformal homeomorphisms, with an upper bound on the maximal distortion of these homeomorphisms. In the setting of hyperbolic surfaces, this turns out to be a very intriguing question.
Our main result states that there exists a universal lower bound on the maximal dilatation of elements of a transitive family of quasiconformal homeomorphisms on a hyperbolic surface of genus zero
The C*-algebra of an affine map on the 3-torus
We study the C*-algebra of an affine map on a compact abelian group and give
necessary and sufficient conditions for strong transitivity when the group is a
torus. The structure of the C*-algebra is completely determined for all
strongly transitive affine maps on a torus of dimension one, two or three
Construction of curious minimal uniquely ergodic homeomorphisms on manifolds: the Denjoy-Rees technique
Mary Rees has constructed a minimal homeomorphism of the 2-torus with
positive topological entropy. This homeomorphism f is obtained by enriching the
dynamics of an irrational rotation R. We improve Rees construction, allowing to
start with any homeomorphism R instead of an irrational rotation and to control
precisely the measurable dynamics of f. This yields in particular the following
result: Any compact manifold of dimension d>1 which carries a minimal uniquely
ergodic homeomorphism also carries a minimal uniquely ergodic homeomorphism
with positive topological entropy. More generally, given some homeomorphism R
of a (compact) manifold and some homeomorphism h of a Cantor set, we construct
a homeomorphism f which "looks like" R from the topological viewpoint and
"looks like" R*h from the measurable viewpoint. This construction can be seen
as a partial answer to the following realisability question: which measurable
dynamical systems are represented by homeomorphisms on manifolds
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