891 research outputs found
Dynamical consequences of a free interval: minimality, transitivity, mixing and topological entropy
We study dynamics of continuous maps on compact metrizable spaces containing
a free interval (i.e., an open subset homeomorphic to an open interval). A
special attention is paid to relationships between topological transitivity,
weak and strong topological mixing, dense periodicity and topological entropy
as well as to the topological structure of minimal sets. In particular, a
trichotomy for minimal sets and a dichotomy for transitive maps are proved.Comment: 21 page
Strong topological transitivity, hypermixing, and their relationships with other dynamical properties
Recently, two stronger versions of dynamical properties have been introduced
and investigated: strong topological transitivity, which is a stronger version
of the topological transitivity property, and hypermixing, which is a stronger
version of the mixing property. We continue the investigation of these notions
with two main results. First, we show there are dynamical systems which are
strongly topologically transitive but not weakly mixing. We then show that on
or , there is a weighted backward shift which is strongly
topologically transitive but not mixing
A Birkhoff type transitivity theorem for non-separable completely metrizable spaces with applications to Linear Dynamics
In this note we prove a Birkhoff type transitivity theorem for continuous
maps acting on non-separable completely metrizable spaces and we give some
applications for dynamics of bounded linear operators acting on complex
Fr\'{e}chet spaces. Among them we show that any positive power and any
unimodular multiple of a topologically transitive linear operator is
topologically transitive, generalizing similar results of S.I. Ansari and F.
Le\'{o}n-Saavedra V. M\"{u}ller for hypercyclic operators.Comment: Several changes concerning the presentation of the paper; title
changed; 12 page
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