Diophantine conditions and real or complex Brjuno functions

The continued fraction expansion of the real number x=a_0+x_0, a_0\in {\ZZ}, is given by 0\leq x_n<1, x_{n}^{-1}=a_{n+1}+ x_{n+1}, a_{n+1}\in {\NN}, for $n\geq 0.$ The Brjuno function is then $B(x)=\sum_{n=0}^{\infty}x_0x_1... x_{n-1}\ln(x_n^{-1}),$ and the number $x$ satisfies the Brjuno diophantine condition whenever $B(x)$ is bounded. Invariant circles under a complex rotation persist when the map is analytically perturbed, if and only if the rotation number satisfies the Brjuno condition, and the same holds for invariant circles in the semi-standard and standard maps cases. In this lecture, we will review some properties of the Brjuno function, and give some generalisations related to familiar diophantine conditions. The Brjuno function is highly singular and takes value $+\infty$ on a dense set including rationals. We present a regularisation leading to a complex function holomorphic in the upper half plane. Its imaginary part tends to the Brjuno function on the real axis, the real part remaining bounded, and we also indicate its transformation under the modular group.Comment: latex jura.tex, 6 files, 19 pages Proceedings on `Noise, Oscillators and Algebraic Randomness' La Chapelle des Bois, France 1999-04-05 1999-04-10 April 5-10, 1999 [SPhT-T99/116

Linearizability of Saturated Polynomials

Brjuno and R\"ussmann proved that every irrationally indifferent fixed point of an analytic function with a Brjuno rotation number is linearizable, and Yoccoz proved that this is sharp for quadratic polynomials. Douady conjectured that this is sharp for all rational functions of degree at least 2, i.e., that non-M\"obius rational functions cannot have Siegel disks with non-Brjuno rotation numbers. We prove that Douady's conjecture holds for the class of polynomials for which the number of infinite tails of critical orbits in the Julia set equals the number of irrationally indifferent cycles. As a corollary, Douady's conjecture holds for the polynomials $P(z) = z^d + c$ for all $d > 1$ and all complex $c$.Comment: 28 pages, major revisions and additions following referee comment

Scaling of the Critical Function for the Standard Map: Some Numerical Results

The behavior of the critical function for the breakdown of the homotopically non-trivial invariant (KAM) curves for the standard map, as the rotation number tends to a rational number, is investigated using a version of Greene's residue criterion. The results are compared to the analogous ones for the radius of convergence of the Lindstedt series, in which case rigorous theorems have been proved. The conjectured interpolation of the critical function in terms of the Bryuno function is discussed.Comment: 26 pages, 3 figures, 13 table