Marco Martens

Distortion results and invariant Cantor sets of unimodal maps

A distortion theory is developed for

The mathematics of halftoning

S-

Universal models for Lorenz maps

unimodal maps. It will be used to get some geometric understanding of invariant Cantor sets. In particular attracting Cantor sets turn out to have Lebesgue measure zero. Furthermore the ergodic behavior of

Red spectra from white and blue noise.

S-

THE MULTIPLIERS OF PERIODIC POINTS IN ONE-DIMENSIONAL DYNAMICS

unimodal maps is classified according to a distortion property, called the Markov-property.

Convex dynamics and applications

This paper describes some mathematical aspects of halftoning in digital printing. Halftoning is the technique of rendering a continuous range of colors using only a few discrete ones. There are two major classes of methods: dithering and error diffusion. Some discussion is presented concerning the method of dithering, but the main emphasis is on error diffusion.

Error bounds for error diffusion and related digital halftoning algorithms

The existence of smooth families of Lorenz maps exhibiting all possible dynamical behavior is established and the structure of the parameter space of these families is described.

Error bounds for error diffusion and other mathematical problems arising in digital halftoning

The value of maps of the interval in modelling population dynamics has recently been called into question because temporal variations from such maps have blue or white power spectra, whereas many observations of real populations show time–series with red spectra. One way to deal with this discrepancy is to introduce chaotic or stochastic fluctuations in the parameters of the map. This leads to on–off intermittency and can markedly redden the spectrum produced by a model that does not by itself have a red spectrum. The parameter fluctuations need not themselves have a red spectrum in order to achieve this effect. Because the power spectrum is not invariant under a change of variable, another way to redden the spectrum is by a suitable transformation of the variables used. The question this poses is whether spectra are the best means of characterizing a fluctuating variable.

Rigidity of critical circle maps

It will be shown that the smooth conjugacy class of an S-unimodal map which has neither a periodic attractor nor a Cantor attractor is determined by the multipliers of the periodic orbits. This generalizes a result by M Shub and D Sullivan (1985 Expanding endomorphism of the circle revisited Ergod. Theor. Dynam. Sys. 5 285-9) for smooth expanding maps of the circle.

Forcing of periodic orbits for interval maps and renormalization of piecewise affine maps

This paper proves a theorem about bounding orbits of a time dependent dynamical system. The maps that are involved are examples in convex dynamics, by which we mean the dynamics of piecewise isometrics where the pieces are convex. The theorem came to the attention of the authors in connection with the problem of digital halftoning. Digital halftoning is a family of printing technologies for getting full-color images from only a few different colors deposited at dots all of the same size. The simplest version consists in obtaining gray-scale images from only black and white dots. A corollary of the theorem is that for error diffusion, one of the methods of digital halftoning, averages of colors of the printed dots converge to averages of the colors taken from the same dots of the actual images. Digital printing is a special case of a much wider class of scheduling problems to which the theorem applies. Convex dynamics has roots in classical areas of mathematics such as symbolic dynamics, Diophantine approximation, and the theory of uniform distributions.