Proof. One can choose a finite number T = T (S) of train tracks τ1,...,τT
such that (1) every curve is carried by τi for some i and (2) the set of curves that
are carried by τi and have positive weights on all branches of τi is disjoint from the
corresponding set of curves carried by τj. Pairs of attracting and repelling fixed
points of pseudo-Anosov diffeomorphisms are dense in PML × PML.
If there is an invariant mean σ on S, for some i the set Bi of curves carried by τi
satisfies ai = σ(Bi) 0. Choose N such that N 1/ai for such an index i. Find N
pseudo-Anosovs ψk with distinct attracting fixed points at laminations carried by
τi and repelling fixed points carried by some τj with j = i. Disjoint neighborhoods
around these attracting fixed points may be chosen such that all curves in each
neighborhood are carried by τi and have positive weights on all branches of τi;
likewise for the repelling fixed points. Raising each ψk to a high enough power mk
we can conclude that the curves carried by ψk
(τi) are disjoint from the set of
curves carried by ψl
(τi) for l = k. By invariance, each of those sets has σ-mean
ai. Adding N of these we find that σ(S) 1, a contradiction.
Corollary 2.4. There is no Mod(S)-invariant Borel probability measure μ on
3. Genericity in the curve complex
In the paper [4], Lustig–Moriah give the following notion of genericity.
Definition 3.1. Let X be a topological space, provided with a Borel measure
or measure class μ. Let Y X be a (possibly countable) subset with μ(Y ) = 0.
Then the set A Y is called generic in Y (or simply generic, if Y = X) if
μ(Y \ A) = 0. (Here closures are taken in X.) On the other hand A is called null
in Y if μ(A) = 0.
We can extend the definitions to products as follows. Given E Y × Y and
a Y let E(a) := {b Y : (a, b) E or (b, a) E}.
Definition 3.2. E is null in Y × Y if {a Y : E(a) not null in Y } is null.
This definition for products corresponds to Fubini’s theorem: the set of points
with non-null fibers must be null.
We will focus on the case that X = PML(S) for any surface S and Y = S(S)
is the set of simple closed curves. Several examples and observations can be made
immediately to illustrate that this notion is topologically interesting.
Nullness in S is preserved by: acting by Mod(S), passing to subsets, and
finite unions. A set is null if and only if its complement is generic.
The entire set S is generic in S, and being generic in S implies denseness
in PML. (Because if A misses an open set in PML, then the closure of
its complement has positive measure.)
There are natural subsets of S that are neither null nor generic. For
instance, suppose that g 2, so that S has a nontrivial partition into
separating and nonseparating curves. Each of these subsets is dense in
PML, so neither can be null or generic.
Our basic example of a null set in S is the set of all curves disjoint
from some α, which is a copy of S(Sα) sitting inside S(S). Its closure
in PML(S) consists of those laminations giving zero weight to α, which
Previous Page Next Page