A simpler derivation of Feigenbaum's renormalization group equation for the period-doubling bifurcation sequence Am.We construct a renormalization fixed point corresponding to the strong coupling limit of the golden mean Harper equation. We give an analytic expression for this fixed point, establish its existence and uniqueness, and verify properties previously seen only in numerical calculations. The spectrum of the linearization of the renormalization operator at this fixed point is also explicitly determined. This strong coupling fixed point also helps describe the onset of a strange nonchaotic attractor in quasiperiodically forced systems.

Abstract. We study the growth at the golden rotation number ω = ( √ 5 − 1)/2 of the function sequence P n (ω) = n r=1 |2 sin πrω|. This sequence has been variously studied elsewhere as a skew product of sines, Birkho sum, q-Pochhammer symbol (on the unit circle), and restricted Euler function. In particular we study the Fibonacci decimation of the sequence P n , namely the sub-sequence Q n = Fn r=1 2 sin πrω for Fibonacci numbers F n , and prove that this renormalisation subsequence converges to a constant. From this we show rigorously that the growth of P n (ω) is bounded by power laws. This provides the theoretical basis to explain recent experimental results reported by Knill and Tangerman (Self-similarity and growth in Birkho sums for the golden rotation. Nonlinearity, 24(11):31153127, 2011).

We calculate all piecewise-constant periodic orbits (with values ±1) of the renormalisation recursion arising in the analysis of correlations of the orbit of a point on a strange nonchaotic attractor. Our results make rigorous and generalise previous numerical results.

We give a rigorous renormalization analysis of the self-similarity of correlation functions in a quasiperiodically forced two-level system. More precisely, the system considered is a quantum two-level system in a time-dependent field consisting of periodic kicks with amplitude given by a discontinuous modulation function driven in a quasiperiodic manner at golden mean frequency. Mathematically, our analysis consists of a description of all piecewise-constant periodic orbits of an additive functional recurrence. We further establish a criterion for such orbits to be globally bounded functions. In a particular example, previously only treated numerically, we further calculate explicitly the asymptotic height of the main peaks in the correlation function.

We provide a rigorous analysis of the fluctuations of localized eigenstates in a generalized Harper equation with golden mean flux and with next-nearest-neighbor interactions. For next-nearest-neighbor interaction above a critical threshold, these self-similar fluctuations are characterized by orbits of a renormalization operator on a universal strange attractor, whose projection was dubbed the "orchid" by Ketoja and Satija [Phys. Rev. Lett. 75, 2762 (1995)]. We show that the attractor is given essentially by an embedding of a subshift of finite type, and give a description of its periodic orbits.