PhD position available in Ergodic Theory at Utrecht University with
title Merging Critical Orbits. The position is sponsored by a grant
from the Dutch National Science Foundation.
General Description of the intended research:
Matching is a mysterious phenomenon which has recently been
observed for several parameterized families of interval maps in the
deterministic and random settings. It is the property that for each
critical point the (random) orbits of the left and right limit merge
after some finite number of steps, and that the (expected value of the)
derivatives of both orbits are also equal at that time; this assures the
stability of this phenomenon under small perturbations of the parameter.
Since most of the dynamical behaviour of systems are encoded in the
possible trajectories of the critical points, knowledge on when and how
matching occurs can help in finding explicit expression for the natural
invariant measure. Once such a measure is found, one is able to obtain
essential information regarding the system, such as the frequency the
orbits enter a specific region, the entropy, the Lyapunov exponents,
mixing rates etc., and to make comparisons as the parameter varies.
There are many theorems that assert the existence of such invariant
measures, but there are few results that give a recipe for an explicit
formula, which is essential in describing the exact asymptotic
behaviour. We propose a new methodology to construct such measures and
to uncover their properties and behaviour under parameterized
perturbations. Our aim is to give a systematic way of analysing such
systems and to provide a new approach, with the help of matching, to
relate non-isomorphic systems that exhibit similar matching phenomena
and to extract information from one system to the other.
