16 research outputs found
Expansive homeomorphisms of the plane
This article tackles the problem of the classification of expansive
homeomorphisms of the plane. Necessary and sufficient conditions for a
homeomorphism to be conjugate to a linear hyperbolic automorphism will be
presented. The techniques involve topological and metric aspects of the plane.
The use of a Lyapunov metric function which defines the same topology as the
one induced by the usual metric but that, in general, is not equivalent to it
is an example of such techniques. The discovery of a hypothesis about the
behavior of Lyapunov functions at infinity allows us to generalize some results
that are valid in the compact context. Additional local properties allow us to
obtain another classification theorem.Comment: 29 pages, 22 figure
Infinitesimal Lyapunov functions for singular flows
We present an extension of the notion of infinitesimal Lyapunov function to
singular flows, and from this technique we deduce a characterization of
partial/sectional hyperbolic sets. In absence of singularities, we can also
characterize uniform hyperbolicity.
These conditions can be expressed using the space derivative DX of the vector
field X together with a field of infinitesimal Lyapunov functions only, and are
reduced to checking that a certain symmetric operator is positive definite at
the tangent space of every point of the trapping region.Comment: 37 pages, 1 figure; corrected the statement of Lemma 2.2 and item (2)
of Theorem 2.7; removed item (5) of Theorem 2.7 and its wrong proof since the
statement of this item was false; corrected items (1) and (2) of Theorem 2.23
and their proofs. Included Example 6 on smooth reduction of families of
quadratic forms. The published version in Math Z journal needs an errat