Let G be the group PAff+(R/Z) of piecewise affine circle homeomorphisms or the group Diff∞(R/Z) of smooth circle diffeomorphisms. A constructive proof that all irrational rotations are distorted in G is given.

We investigate the curves in the complex plane which are generated by sequences of real numbers being the lifts of the points on the orbit of an orientation preserving circle homeomorphism. Geometrical properties of these curves such as boundedness, superficiality, local discrete radius of curvature are linked with dynamical properties of the circle homeomorphism which generates them: rotation number and its continued fraction...

The dataset entitled Computing dynamical curlicues contains values of consecutive points on a curlicue generated, respectively, by rotation on the circle by different angles, the Arnold circle map (with various parameter values) and an exemplary sequence as well as corresponding diameters and Birkhoff averages of these curves. We additionally provide source codes of the Matlab programs which can be used to generate and plot the...

The central limit theorem and law of the iterated logarithm for Markov chains corresponding to random walks on the space Homeo(S1) of circle homeomorphisms for centered Lipschitz functions and every starting point are proved.

We consider the Leaky Integrate-and-Fire and Perfect Integrator models of neuron’s dynamics with the input function being periodic and almost-periodic (in the sense of Stepanov). In particular we analyze properties and dynamics of the so-called firing map, which iterations give timings of consecutive spikes of a neuron. In case of a periodic input function we provide a detailed description of the sequence of interspike-intervals,...