On the Regularity of the Displacement Sequence of an Orientation Preserving Circle Homeomorphism (original) (raw)

We investigate the regularity properties of the displacement sequence () () () (), 2 exp , 1 mod 1 ix z x x z n n n π = Φ − Φ = η − where R R → Φ : is a lift of an orientation preserving circle homeomorphism. If the rotation number () q p = ϕ is rational, then () z n η is asymptotically periodic with semi-period q. This WACŁAW MARZANTOWICZ and JUSTYNA SIGNERSKA 12 convergence to a periodic sequence is uniform in z if we admit that some points are iterated backward instead of taking only forward iterations for all z. This leads to the notion of an basins'-ε edge, which we illustrate by the numerical example. If () , Q ∈ / ϕ then some classical results in topological dynamics yield that the displacement sequence also exhibits some regularity properties, which we define and prove in the second part of the paper.