Combinatorics of distance doubling maps ∗

Combination laws for scaling exponents and relation to the geometry of renormalization operators

Renormalization group has become a standard tool for describing universal properties of different routes to chaos – period-doubling in unimodal maps, quasiperiodic transitions in circle maps, dynamics on the boundaries of Siegel disks, destruction of invariant circles of area-preserving twist maps, and others. The universal scaling exponents for each route are related to the properties of the c...

Renormalization Ideas in Conformal Dynamics

1. Introduction How to look at a dynamical system f at a small scale? You should take a small piece of the phase space, consider the rst return map to this piece, and then rescale it to \the original size". The new dynamical system is called the renormalization Rf of the original one. It may happen that Rf looks \similar" to f, and then you can try to repeat this procedure, and construct the se...

Renormalization Ideas in Conformal Dynamics Mikhail Lyubich

How to look at a dynamical system f at a small scale You should take a small piece of the phase space consider the rst return map to this piece and then rescale it to the original size The new dynamical system is called the renormalization Rf of the original one It may happen that Rf looks similar to f and then you can try to repeat this procedure and construct the second renormalization R f et...

Doubling and Tripling Constructions for Defining Sets in Steiner Triple Systems

Bifurcations of beam – beam like maps

The bifurcations of a class of mappings including the beam–beam map are examined. These maps are asymptotically linear at infinity where they exhibit invariant curves and elliptic periodic points. The dynamical behaviour is radically different with respect to the Hénon-like polynomial maps whose stability boundary (dynamic aperture) is at a finite distance. Rather than the period-doubling bifur...