Multifractal Spectrum of an Experimental (Video Feedback) Farey Tree

Abstract

A camera films a screen to which it is connected. It films its own image, feeding back the image to the screen. The camera can turn around an optical axis. A pattern of p light spots on the screen and q turns (feedback loops) of the camera appears, where p and q follow the hierarchy of a Farey tree. The Farey tree induces a measure distribution μ on the unit segment, different from the hyperbolic one μH induced by the Farey-Brocot interpolation. In this paper the multifractal spectrum of μ is studied and compared with that of μH; the study of the latter spectrum is refined. The spectra are studied in this paper by means of different tools from Number Theory. The results of this study are interpreted in terms of p and q, empirically obtained in the video feedback experiment.