  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.
