Anders Öberg, Uppsala University
In this talk I present first a recent work with Anders Johansson and Mark Pollicott, in which we prove that there exists a continuous eigenfunction for the transfer operator corresponding to potentials for the classical Dyson model in the subcritical regime when the parameter alpha is greater than 3/2. This is a significant improvement on previous results where the existence of a continuous eigenfunction of the transfer operator was only established for general potentials satisfying summable variations, which would correspond to the parameter range alpha > 2.
In the second half of the talk I present a recent result with Noam Berger, Diana Conache and Anders Johansson, in which we prove that uniqueness of a Doeblin measure, a g-measure, follows if the variations of the specification is less than 2/square root of n, where we think that 2 may be a sharp constant. We also give an example of a unique Doeblin measure which is not (weak) mixing, hence the sequence of the iterates of the transfer operator does not converge.
The two parts of the talk are connected since the “lack of the g-measure property” in the supercritical regime of the Dyson model implies that there is no continuous eigenfunction of the transfer operator.