Irrational rotation

Abstract: This paper presents the first examples of K3 surface automorphisms \(f : X \rightarrow X\) with Siegel disks (domains on which f acts by an irrational rotation). The set of such examples is countable, and the surface \(X\) must be non-projective to carry a Siegel disk. These automorphisms are synthesized from Salem numbers of degree 22 and trace −1, which play the role of the leading eigenvalue for \(f*|H^2(X)\). The construction uses the Torelli theorem, the Atiyah-Bott fixed-point theorem and results from transcendence theory.

Abstract: Let Ca be the central Cantor set obtained by removing a central interval of length 1−2a from the unit interval, and then continuing this process inductively on each of the remaining two intervals. We prove that if log b/log a is irrational, then where dim is Hausdorff dimension. More generally, given two self-similar sets K,K′ in ℝ and a scaling parameter s>0, if the dimension of the arithmetic sum K+sK′ is strictly smaller than dim (K)+dim (K′)≤1 (‘geometric resonance’), then there exists r<1 such that all contraction ratios of the similitudes defining K and K′ are powers of r (‘algebraic resonance’). Our method also yields a new result on the projections of planar self-similar sets generated by an iterated function system that includes a scaled irrational rotation.

Abstract: We consider families of maps of the circle of degree 1 which are homeomorphisms but not diffeomorphisms, that is maps like $$x \to x + t + \frac{c}{{2\pi }}\sin (2\pi x)(\bmod 1)$$ withc=1. We prove that the set of parameter values corresponding to irrational rotation numbers has Lebesgue measure 0. In other words, the intervals on which frequency-locking occurs fill up the set of full measure.

Abstract: We prove that two C 3 critical circle maps with the same rotation number in a special set ? are C 1+α conjugate for some α>0 provided their successive renormalizations converge together at an exponential rate in the C 0 sense. The set ? has full Lebesgue measure and contains all rotation numbers of bounded type. By contrast, we also give examples of C ∞ critical circle maps with the same rotation number that are not C 1+β conjugate for any β>0. The class of rotation numbers for which such examples exist contains Diophantine numbers.