Thierry Giordano, University of Ottawa
"Topological orbit equivalence: an overview!"
In
1959,
H.
Dye
introduced
the
notion
of
orbit
equivalence
and
proved
that
any
two
ergodic
finite
measure
preserving
transformations
on
a
Lebesgue
space
are
orbit
equivalent.
He
also
conjectured
that
an
arbitrary
action
of
a
discrete
amenable
group
is
orbit
equivalent
to
a
Z-action.
This
conjecture
was
proved
by
Ornstein
and
Weiss
and
its
most
general
case
by
Connes,
Feldman
and
Weiss
by
establishing
that
an
amenable
non-singular
countable
equivalence
relation
R
can
be
generated
by
a
single
transformation,
or
equivalently
is
hyperfinite,
i.e.,
R
is
up
to
a
null
set,
a
countable
increasing
union
of
finite
equivalence
relations.
In
1992,
using
ideas
of
A.
Vershik,
Richard
Herman,
Ian
Putnam
and
Christian
Skau
constructed
a
remarkable
model
for
minimal
homeomorphisms
of
the
Cantor
set.
They
associated
to
a
Cantor
min-
imal
system
(X,φ)
an
ordered
Bratteli
diagram
and
proved
that
the
corresponding
Bratteli-Vershik
transformation
is
conjugate
to
φ.
The
approximately
finite
(AF)
equivalence
relation
and
the
dimen-
sion
groups
associated
to
this
model
were
key
in
the
classification
up
to
orbit
equivalence
of
minimal
free
actions
of
Z,
and
later
of
Zd.
In
this
talk
I
will
review
these
results
of
topological
orbit
equivalence
and
present
a
generalization
of
the
Bratteli-Vershik
model
for
some
minimal
free
actions
of
Z2
on
the
Cantor
set.