Rational Parameter Rays of The Multibrot Sets

We prove a structure theorem for the multibrot sets, which are the higher degree analogues of the Mandelbrot set, and give a complete picture of the landing behavior of the rational parameter rays and the bifurcation phenomenon. Our proof is inspired by previous works of Schleicher and Milnor on the combinatorics of the Mandelbrot set; in particular, we make essential use of combinatorial tools such as orbit portraits and kneading sequences. However, we avoid the standard global counting arguments in our proof and replace them by local analytic arguments to show that the parabolic and the Misiurewicz parameters are landing points of rational parameter rays

The Relation Between KMS-states for Different Temperatures

Given a thermal field theory for some temperature $\beta^{-1}$, we construct the theory at an arbitrary temperature $1 / \beta'$. Our work is based on a construction invented by Buchholz and Junglas, which we adapt to thermal field theories. In a first step we construct states which closely resemble KMS states for the new temperature in a local region \O_\circ \subset \rr^4, but coincide with the given KMS state in the space-like complement of a slightly larger region \hat{\O}. By a weak*-compactness argument there always exists a convergent subnet of states as the size of \O_\circ and \hat{\O} tends towards \rr^4. Whether or not such a limit state is a global KMS state for the new temperature, depends on the surface energy contained in the layer in between the boundaries of \O_\circ and \hat{\O}. We show that this surface energy can be controlled by a generalized cluster condition.Comment: latex, 24 page