🚀 go-pugleaf

I am considering the problem of expanding maps from the circle to itself.
For f to be expanding, I mean that f is C^1 and |f'(x)| >= K >1 for all x
in S^1. Such maps may be "lifted" to the interval, where they become
piecewise expanding, once-differentiable maps of the interval, whose
branches map onto the whole interval.

If the map f is C^2 or even C^(1+\alpha) (i.e. its derivative is Hoelder
continuous), then it is known, using the theory of Lasota-Yorke and others
that the map has an Absolutely Continuous Invariant Probability Measure
(ACIPM) (with respect to Lebesgue measure) and that this measure is unique.

In the case where the map is C^1, there need not be an ACIPM, as shown
in the paper of Gora and Schmitt in Ergodic Theory and Dynamical Systems
(1988?).

I am interested in the case where the map is C^1 and preserves Lebsegue
measure. The question which I want to answer is whether this probability
measure is unique (i.e. whether in that case, Lebsegue measure is ergodic).

I have some partial results. These are the following:
For C^0 maps, expanding can be defined that all sufficiently small distances
are expanded by a factor uniformly bounded below by some K>1. One can then
ask if a C^0 expanding map of the circle which preserves Lebesgue measure
can have more than 1 ACIPM. The answer to this is yes, it may have more
than 1 ACIPM. My example has a Cantor set C of positive measure such that
f(C) is contained in C. I have shown that this cannot arise for C^1 maps.
I would be happy to show the proofs to anyone interested.

I should be extremely interested to hear of any comments or suggestions
which people may have with this problem, as I am fairly stuck with it.

Thanks in advance!

Anthony Quas                     anq@maths.warwick.ac.uk
Maths Institute
University of Warwick
Coventry CV4 7AL
ENGLAND