Distribution of the displacement sequence of an orientation preserving circle homeomorphism

Abstract

We give a complete description of the behaviour of the sequence of displacements η n (z) = Φ n (x)−Φ n−1 (x) mod 1, z = exp(2πix), along a trajectory {ϕ n (z)}, where ϕ is an orientation preserving circle homeomorphism and Φ : R → R its lift. If the rotation number ̺(ϕ) = p q is rational then η n (z) is asymptotically periodic with semi-period q. This 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. If ̺(ϕ) / ∈ Q then the values of η n (z) are dense in a set which depends on the map γ (semi-)conjugating ϕ with the rotation by ̺(ϕ) and which is the support of the displacements distribution. We provide an effective formula for the displacement distribution if ϕ is C 1 -diffeomorphism and show approximation of the displacement distribution by sample displacements measured along a trajectory of any other circle homeomorphism which is sufficiently close to the initial homeomorphism ϕ. Finally, we prove that even for the irrational rotation number ̺ the displacement sequence exhibits some regularity properties.

Analysis of Interspike-Intervals for the General Class of Integrate-and-Fire Models With Periodic Drive

“…Now this result is a direct consequence of Theorem 2.17 in [19] about approximation in the Fortet-Mourier metric by the sample displacement distribution of a homeomorphism which is just close enough to the given homeomorphism.…”

“…Since the firing phase map ϕ is a circle homeomorphism with the irrational rotation number, the statement follows from the fact that every point z ∈ S 1 for ϕ transitive (similarly, every point z ∈ ∆ for non-transitive case) is almost periodic under the dynamical system (ϕ, S 1 ). This was shown in [19] basing on the fact (the classical result proved in [13]) that every point x ∈ X is almost-periodic under ϕ, if X is a compact metric space and X is minimal for ϕ (the set is minimal if it is non-empty, closed, invariant and such that no proper subset of it shares all these properties).…”

“…In [19] we proved many properties of the displacement distribution µ Ψ of an orientation preserving circle homeomorphism ϕ with an irrational rotation number which is defined exactly as in Definition 2, up to taking mod 1. Thus interspike-interval distribution µ ISI of the firing map Φ has exactly the same properties as the displacement distribution µ Ψ .…”

“…We recall that even if the invariant measure µ, which gives the distribution of firing phases, is absolutely continuous, it might not be so for the interspike-interval distribution µ ISI as happens for example for the systemẋ =F (x) + I with constant input I, where the firing map Φ is a lift of the rotation and the interspike-interval distribution is Dirac delta centered at , being the rotation number of Φ (and thus µ ISI is singular). We formulate the following theorem which is just a paraphrase of its correspondence for the displacement sequence of a circle homeomorphism (diffeomorphism), Theorem 2.14 in [19], and thus will be stated without the proof:…”

“…Then assuming that F is also periodic in t, we give a detailed description of the sequence of interspike-intervals and interspike-interval distribution. Here we make use of mathematical result concerning displacement sequence of an orientation preserving circle homeomorphism proved in [19] and [24].…”