59,855 research outputs found
On the omega-limit sets of tent maps
For a continuous map f on a compact metric space (X,d), a subset D of X is
internally chain transitive if for every x and y in D and every delta > 0 there
is a sequence of points {x=x_0,x_1, ...,x_n=y} such that d(f(x_i),x_{i+1}) <
delta for i=0,1, ...,n-1. It is known that every omega-limit set is internally
chain transitive; in earlier work it was shown that for X a shift of finite
type, a closed subset D of X is internally chain transitive if and only if D is
an omega-limit set for some point in X, and that the same is also true for the
tent map with slope equal to 2. In this paper, we prove that for tent maps
whose critical point c=1/2 is periodic, every closed, internally chain
transitive set is necessarily an omega-limit set. Furthermore, we show that
there are at least countably many tent maps with non-recurrent critical point
for which there is a closed, internally chain transitive set which is not an
omega-limit set. Together, these results lead us to conjecture that for those
tent maps with shadowing (or pseudo-orbit tracing), the omega-limit sets are
precisely those sets having internal chain transitivity.Comment: 17 page
Quasi-pseudo-metrization of topological preordered spaces
We establish that every second countable completely regularly preordered
space (E,T,\leq) is quasi-pseudo-metrizable, in the sense that there is a
quasi-pseudo-metric p on E for which the pseudo-metric p\veep^-1 induces T and
the graph of \leq is exactly the set {(x,y): p(x,y)=0}. In the ordered case it
is proved that these spaces can be characterized as being order homeomorphic to
subspaces of the ordered Hilbert cube. The connection with
quasi-pseudo-metrization results obtained in bitopology is clarified. In
particular, strictly quasi-pseudometrizable ordered spaces are characterized as
being order homeomorphic to order subspaces of the ordered Hilbert cube.Comment: Latex2e, 20 pages. v2: minor changes in the proof of theorem 2.
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