1.4. Diffeomorphisms with nonzero Lyapunov exponents on surfaces 25
The factor-space
T2/I
is homeomorphic to the sphere
S2
(see Figure 1.7)
and admits a natural smooth structure induced from the torus everywhere
except for the points xi, i = 1,2,3,4. Moreover, the following statement
holds.
Proposition 1.23 (Katok [45]). There exists a map ζ :
T2

S2
satisfying:
(1) ζ is a double branched covering, is regular (one-to-one on each
branch), and is
C∞
everywhere except at the points xi, i = 1,2,3,4,
where it branches;
(2) ζ I = ζ;
(3) zeta preserves area; i.e., ζ∗m = μ where μ is the area in
S2;
(4) there exists a local coordinate system in a neighborhood of each point
ζ(xi), i = 1,2,3,4, in which
ζ(s1,s2) =
s
2

s22
√1
s12 + s22
,
2s1s2

s12 + s22
in each disk
Dr1.i
Finally, one can show that the map GS2 = ζ G2
ζ−1
is a
C∞
diffeo-
morphism which preserves area. It is easy to see that this map has nonzero
Lyapunov exponents almost everywhere. It is also ergodic (and, indeed, is
a Bernoulli diffeomorphism).
The diffeomorphism GS2 can be used to build an area-preserving
C∞
diffeomorphism with nonzero Lyapunov exponents on any surface. This is a
two step procedure. First, the sphere can be unfolded into the unit disk and
the map GS2 can be carried over to a
C∞
area-preserving diffeomorphism
g of the disk. The second step is to cut the surface in a certain way so that
the resulting surface with the boundary is diffeomorphic to the disk. This
is a well-known topological construction but we need to complement it by
showing that the map g can be carried over to produce the desired map
on the surface. In doing so, a crucial fact is that g is the identity on the
boundary of the disk and is “sufficiently flat” near the boundary. We shall
briefly outline the procedure without going into details.
Set pi = ζ(xi), i = 1,2,3,4. In a small neighborhood of the point p4 we
define a map η by
η(τ1,τ2) =
τ1 1 τ1
2
τ2
2
τ1 2 + τ2 2
,
τ2 1 τ1
2
τ2
2
τ1 2 + τ22
.
One can extend it to an area-preserving
C∞
diffeomorphism η between
S2
\
{p4} and the unit disk
D2

R2.
The map
g = η GS2
η−1
(1.15)
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