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)