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Robust transitivity and topological mixing for -flows
We prove that non-trivial homoclinic classes of -generic flows are
topologically mixing. This implies that given a non-trivial
-robustly transitive set of a vector field , there is a
-perturbation of such that the continuation of
is a topologically mixing set for . In particular, robustly
transitive flows become topologically mixing after -perturbations. These
results generalize a theorem by Bowen on the basic sets of generic Axiom A
flows. We also show that the set of flows whose non-trivial homoclinic classes
are topologically mixing is \emph{not} open and dense, in general.Comment: Final version, to appear in the Proceedings of the AM
James bundles
We study cubical sets without degeneracies, which we call {square}-sets. These sets arise naturally in a number of settings and they have a beautiful intrinsic geometry; in particular a {square}-set C has an infinite family of associated {square}-sets Ji(C), for i = 1, 2, ..., which we call James complexes. There are mock bundle projections pi: |Ji(C)| -> |C| (which we call James bundles) defining classes in unstable cohomotopy which generalise the classical JamesâHopf invariants of {Omega}(S2). The algebra of these classes mimics the algebra of the cohomotopy of {Omega}(S2) and the reduction to cohomology defines a sequence of natural characteristic classes for a {square}-set. An associated map to BO leads to a generalised cohomology theory with geometric interpretation similar to that for Mahowald orientation
Equivalence of operations with respect to discriminator clones
For each clone C on a set A there is an associated equivalence relation,
called C-equivalence, on the set of all operations on A, which relates two
operations iff each one is a substitution instance of the other using
operations from C. In this paper we prove that if C is a discriminator clone on
a finite set, then there are only finitely many C-equivalence classes.
Moreover, we show that the smallest discriminator clone is minimal with respect
to this finiteness property. For discriminator clones of Boolean functions we
explicitly describe the associated equivalence relations.Comment: 17 page
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