Institute for Mathematical Sciences
Bjorn Winckler and Marco Martens
Physical Measures for Infinitely Renormalizable Lorenz Maps
Abstract: A physical measure on the attractor of a system describes the statistical behavior of typical orbits. An example occurs
in unimodal dynamics. Namely, all infinitely renormalizable
unimodal maps have a physical measure. For Lorenz dynamics,
even in the simple case of infinitely renormalizable systems,
the existence of physical measures is more delicate.
In this article we construct examples of infinitely renormalizable
Lorenz maps which do not have a physical measure. A priori bounds
on the geometry play a crucial role in (unimodal) dynamics.
There are infinitely renormalizable Lorenz maps which do not
have a priori bounds. This phenomenon is related to the position
of the critical point of the consecutive renormalizations.
The crucial technical ingredient used to obtain these examples
without a physical measure, is the control of the position of
these critical points.
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