Abstract.
In this talk I will survey some developments in ergodic theory which I
have been involved in. The talk will be easily accessible to everyone.
In 1972 (Invent. Math.) Michael Keane introduced $g$-measures into ergodic
theory. These are invariant measures for a map $T$ with a probability
function, a $g$-function, which is a transition probability for the local
inverses of $T$. Finding regularity conditions on the $g$-functions, to
ensure uniqueness of $g$-measures, is a central problem. Keane's results
were extended by Peter Walters in the 1970s and Walters used summability
of
variations of the $g$-functions to ensure uniqueness of $g$-measures. In
the eighties more complicated conditions were used by Berbee, but these
are
still in the realm of $\ell^1$. Such conditions had also been available in
the Markov chain literature, already in the 1950s. But there had also been
mistakes: it had been claimed (in similar contexts) that uniqueness holds
for transition probabilities which are only continuous, but strictly
positive.
In 1993 (Israel J. Math.) Bramson and Kalikow provided a first
counterexample to uniqueness when the $g$-functions are only continuous
(but strictly
positive). In 2003 (Math. Res. Lett.) Anders Johansson and I proved that
square summability of variations of the $g$-functions is enough for
uniqueness
of $g$-measures, and now Berger, Hoffman and Sidoravicius (preprint 2004)
have proved that this condition is also necessary in a certain sense: For
any
$\epsilon>0$ there exists $g$-functions whose variations are in
$\ell^{2+\epsilon}$ and for which there are two distinct $g$-measures.
Recently, Johansson, \"Oberg and Pollicott proved that the condition of
square summability of variations of the $g$-functions can be generalized
in a
natural way to ensure uniqueness of $g$-measures for countable state
shifts.