3 research outputs found
Classification of one dimensional dynamical systems by countable structures
We study the complexity of the classification problem of conjugacy on
dynamical systems on some compact metrizable spaces. Especially we prove that
the conjugacy equivalence relation of interval dynamical systems is Borel
bireducible to isomorphism equivalence relation of countable graphs. This
solves a special case of the Hjorth's conjecture which states that every orbit
equivalence relation induced by a continuous action of the group of all
homeomorphisms of the closed unit interval is classifiable by countable
structures. We also prove that conjugacy equivalence relation of Hilbert cube
homeomorphisms is Borel bireducible to the universal orbit equivalence
relation
The complexity of topological conjugacy of pointed Cantor minimal systems
In this paper, we analyze the complexity of topological conjugacy of pointed
Cantor minimal systems from the point of view of descriptive set theory. We
prove that the topological conjugacy relation on pointed Cantor minimal systems
is Borel bireducible with the Borel equivalence relation
on defined by . Moreover, we show that is a lower bound
for the Borel complexity of topological conjugacy of Cantor minimal systems.
Finally, we interpret our results in terms of properly ordered Bratteli
diagrams and discuss some applications
The complexity of topological conjugacy of pointed Cantor minimal systems
In this paper, we analyze the complexity of topological conjugacy of pointed Cantor minimal systems from the point of view of descriptive set theory. We prove that the topological conjugacy relation on pointed Cantor minimal systems is Borel bireducible with the Borel equivalence relation Delta(+)(R) on R-N defined by x Delta(+)(R)y double left right arrow {x(i):i is an element of N} = {y(i):i is an element of N}. Moreover, we show that Delta(+)(R) is a lower bound for the Borel complexity of topological conjugacy of Cantor minimal systems. Finally, we interpret our results in terms of properly ordered Bratteli diagrams and discuss some applications