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

Surgery on Herman rings of the complex standard family

We consider the standard family (or Arnold family) of circle maps given by f_{\alpha, \beta}(x)=x + \alpha + \beta \sin(x) \pmod{2\pi}, for x,\alpha\in [0,2\pi), \beta \in (0,1) and its complexification F_{\alpha,\beta}(z)=z e^{i\alpha} \exp [\frac12\beta(z-\frac{1}{z})]. If f_{\alpha,\beta} is analytically linearizable, there is a Herman ring around the unit circle in the dynamical plane of F_{\alpha,\beta}. Given an irrational rotation number \theta, the parameters (\alpha,\beta) such that f_{\alpha, \beta} has rotation number \theta form a curve T_\theta in the parameter plane. Using quasi-conformal surgery of the simplest type, we show that if \theta is a Brjuno number, the curve T_\theta can be parametrized real-analytically by the modulus of the Herman ring, from \beta=0 up to a point (\alpha_0,\beta_0) with \beta_0 \leq 1, for which the Herman ring collapses. Using a result of Herman and a construction in I. N. Baker and P. Domínguez (Complex Variables37 (1998), 67-98) we show that for a certain set of angles \theta \in \mathcal{B} \setminus \mathcal{H}, the point \beta_0 is strictly less than 1 and, moreover, the boundary of the Herman rings with the corresponding rotation number have two connected components which are quasi-circles, and do not contain any critical point. For rotation numbers of constant type, the boundary consists of two quasi-circles, each containing one of the two critical points of F_{\alpha, \beta}

Selective photodissociation of tailored molecular tags as a tool for quantum optics

Recent progress in synthetic chemistry and molecular quantum optics has enabled demonstrations of the quantum mechanical wave–particle duality for complex particles, with masses exceeding 10 kDa. Future experiments with even larger objects will require new optical preparation and manipulation methods that shall profit from the possibility to cleave a well-defined molecular tag from a larger parent molecule. Here we present the design and synthesis of two model compounds as well as evidence for the photoinduced beam depletion in high vacuum in one case

Silicon quantum dot devices with a self-aligned second gate layer

We implement silicon quantum dot devices with two layers of gate electrodes using a self-alignment technique, which allows for ultra-small gate lengths and intrinsically perfect layer-to-layer alignment. In a double quantum dot system, we investigate hole transport and observe current rectification due to Pauli spin blockade. Magnetic field measurements indicate that hole spin relaxation is dominated by spin-orbit interaction, and enable us to determine the effective hole $g$-factor $\simeq1.6$. From an avoided singlet-triplet crossing, occurring at high magnetic field, the spin-orbit coupling strength $\simeq0.27$meV is obtained, promising fast and all-electrical spin control