Discrete Holomorphic Local Dynamical Systems

Dynamics of Rational Surface Automorphisms 61
Proof. This is a consequence of the previous Theorem. ⊓⊔
A point p is periodic if f N p = p for some N. The minimal such N > 0 is called
the period of p.
Proposition 1.5. For each N there are only finitely many periodic points of
period N.
Proof. The set PN := {(x,y) ∈ C 2 : f N (x,y) =(x,y)} is a subvariety. Further,
we have PN ⊂ K, so it is bounded. But any bounded subvariety of C 2 must be a
finite set. ⊓⊔
A periodic point p of period N is a saddle if DfN p has one eigenvalue with modulus
1. We use the notation SPer for the set of
saddle (periodic) points and J∗ = SPer for its closure. By an exercise above, we
have:
Proposition 1.6. If p is a saddle (periodic) point, then p ∈ J. Thus J ∗ ⊂ J.
Problem: It is an interesting and basic question to determine whether J∗ = J holds
for all Hénon maps. However, this is not yet known.
Exercise: Suppose that p is a fixed point, and there is a neighborhood U of p such
that fU is a compact subset of U. Show that p is a sink, and � f nU = p.
Theorem 1.7. If |a| = 1, then the volume of (K + ∪ K − ) − Kiszero.If|a| < 1, then
the volume of K − is zero.
Proof. By the previous Corollary, the sets Sn = K − − f n V + are increasing in n.But
the volume is |Sn+1| = |a| 2 |Sn|.If|a|≤1, we have |Sn+1| = |Sn|,or|Sn+1 − Sn| = 0.
Their union is K − −V + = � Sn, so we see that |K − −V + | = 0. By Theorem, then,
we have |K − − K| = 0. If |a| = 1, then we apply the argument to f −1 to obtain the
first statement of the Theorem. If |a| < 1, then the volumes are |K| = | fK| = |a| 2 |K|,
so |K| = 0. ⊓⊔
Problem: Suppose that f is a Hénon map with real coefficients, so f is a diffeomorphism
of R 2 .Ifδ = ±1, then f preserves area. Is it possible for K to have positive
volume in C 2 ?OrifK ⊂ R 2 can it have positive area?
For a point p, wedefinetheω-limit set ω(p) to be the set of all limit points
f n j p → q for subsequences n j → +∞. A property of ω(p) is that it is invariant
under both f and f −1 .
For a compact set X, define its stable set as
W s (X)={y : lim
n→∞ dist( f n y, f n X)=0},
and the unstable set is defined to be the stable set in backward time:
W u (X)={y : lim
n→−∞ dist( f −n y, f −n X)=0}.