Hyperbolicity and Quasi-hyperbolicity in Polynomial Diffeomorphisms of ${\Bbb C}^2$

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by  Eric Bedford

Hyperbolicity and Quasi-hyperbolicity
in Polynomial Diffeomorphisms of C2
Eric Bedford and John Smillie
§0. Introduction. The iteration of holomorphic mappings may be approached from the
6 dynamical and the analytic/geometric points of view. One of the challenges of holomorphic
1 dynamics is to clarify the interplay between these points of view. Analytically we can look, for
0
instance, where the iterates {fn : n ≥ 0} are (or are not) locally equicontinuous. Dynamically,
2
wecanlookforexpansion/contractionalongorbits. ByFriedland-Milnor[FM]weknowthatthe
n
a dynamically interesting polynomial diffeomorphisms of C2 are given by the finite compositions
J of complex H´enon mappings.

3 We start with thebasic dichotomy into bounded/unboundedorbits; K± are the sets where
2
the forward/backward orbits are bounded. Thus K± and K := K+∩K− are the analogues of

] the filled Julia sets for polynomial maps of C. The sets where the forward/backward iterates
S
are not equicontinuous are given by J± := ∂K±. These are the analogues of the Julia set in
D
C and are the complements of the sets of Lyapunov stability in forward/backward time. The
.
h chaotic dynamics takes place inside the set J := J+ ∩J−.
t
a Another analogue of the Julia set in dimension two is the boundary J∗ := ∂ K ⊂ J,
m S
where ∂ denotes the Shilov boundary (in the sense of function algebras). In [BS3] we showed
S
[
that J∗ is equal to the closure of the set S of saddle periodic points. Additional dynamical

1 properties/characterizations for J∗ were obtained in [BS1,3] and [BLS].
v
There are several reasons to be interested in hyperbolic polynomial diffeomorphisms. Our
8
6 understanding of chaotic dynamical systems is most complete in the hyperbolic case. It is
2
also interesting to know how the locus of hyperbolic maps sits in the parameter space and
6
0 the relation between hyperbolicity and structural stability (see [DL] and [BD]). We take hy-
. perbolicity to mean that the system is hyperbolic on its chain recurrent set. For polynomial
1
0 diffeomorphisms this is equivalent to the condition that the set J is a hyperbolic set, though
6 the chain recurrent set may be larger than J, (see [BS1]). Further, it was shown that J = J∗
1
for hyperbolic maps, and the stable manifolds Ws give a Riemann surface lamination of J+. In
:
v
fact, [BS8, Theorem 8.3] characterized hyperbolicity onJ in termsof theexistenceof transverse
i
X Riemann surface laminations of J+ and J− in a neighborhood of J. We will strengthen this
r characterization in Theorem 4.4.
a
Hyperbolicityinvolvesuniformexpansionandcontraction,aswellastransversalitybetween
expanding and contracting directions. In [BS8] we defined a canonical metric k k# on the
q
unstable space Eu ⊂ T C2 for each saddle q ∈ S. A map is said to be quasi expanding if Df
q q
expandsthis metric uniformly (independentlyof q ∈ S) in thestrong sense that there is a κ > 1
with ||Df v||# ≥ κ||v||# for any nonzero v ∈ Eu. For quasi expanding maps, this extends
q f(q) q q
to x ∈ J∗. In [BS8] it was shown that every hyperbolic map is quasi expanding, with k·k#
equivalent to the Euclidean norm. A map is quasi contracting if its inverse is quasi expanding.
We let B(q,r) denote the ball of radius r centered at q and let Ws denote the connected
q,r
component of Ws∩B(q,r) containing x. A geometric characterization of quasi expansion is the
q
Proper, Bounded Area Condition: there exists r > 0 such that for all saddles q ∈ S, (i) Wu
q,r
is proper, i.e., closed in B(q,r), and (ii) the area of Wu is uniformly bounded. This means
q,r
that the degree of local folding of the manifolds {Wu : q ∈ S} near a saddle point q will be
q 0
1
bounded. On the other hand, if {Wu : q ∈ S} if is part of a lamination, then there is no local
q
folding.
Quasi expansion may be viewed as a 2-dimensional analogue of semi-hyperbolicity for
polynomial maps of C (see the Appendix of [BS8]). An important motivation for us was the
work of Carleson, Jones and Yoccoz [CJY], who showed that semi-hyperbolicity was equivalent
to a number of geometric and potential-theoretic properties.
Amapwillbesaidtobequasi-hyperbolicifitisbothquasiexpandingandquasicontracting
(with no requirement of transversality between stable and unstable directions). Quasi hyper-
bolic maps share many properties with hyperbolic ones, and we use this to apply hyperbolic
methods in more general contexts. In the non-hyperbolic case, the canonical metric k k# may
not equivalent to the euclidean metric on C2. However there is a useful filtration of J∗ by a
finitenumberof sets J∗ , each of which carries a metric k k#,ms,mu. This metric k k#,ms,mu
ms,mu x
is equivalent to Euclidean metric for x ∈ J , but it blows up as x approaches a stratum
ms,mu
with larger values of (ms,mu). For maximal values of (ms,mu), J∗ is compact and a
ms,mu
uniformly hyperbolic set.
In Theorem 1 we represent stable manifolds as holomorphic mappings ξs : C → C2, and
x
we write Ws := ξs(C), with Ws := {Ws : x ∈ J∗}. By part (ii) of Theorem 1, {Ws : q ∈ S}
x x x q
extends continuously to Ws. Each Ws is contained in the stable set
x
Ws := {z ∈ C2 : lim dist(fn(x),fn(z)) = 0}
x
n→∞
which, dynamically, is the attracting basin of x. Because of Theorem 3.3 and Corollary 3.4, we
call Ws a stable manifold (as is done in [BD]) even though we do not know yet whether Ws
x x
always coincides with Ws.
x
Theorem 1. Suppose that f is quasi-contracting. With the notation above, we have
(i) For each x ∈ J∗, there is an injective holomorphic immersion ξs : C → J+ ⊂ C2 such that
x
Ws ⊂ Ws.
x x
(ii) There exists r > 0 such that if 0 < r < r , then Ws is a (closed) subvariety of B(x,r),
0 0 x,r
and thedependenceof theclosures J∗ ∋ x 7→ Ws is continuous in theHausdorff topology.
x,r
If f is quasi-expanding, then Theorem 1 holds for ξu and Wu. We write Ws/u for the
x x
families of the stable/unstable manifolds. These stable/unstable manifolds are smooth so it
makes sense to say that they have transverse or tangential intersection. With this, we may
characterize uniform hyperbolicity.
Theorem 2. Suppose that f is quasi-hyperbolic. Then f is uniformly hyperbolic on J∗ if and
only if there is no tangency between Ws and Wu.
Theorem 2 was proved earlier in [BSr] for real H´enon maps of maximal entropy. This
condition is equivalent to J ⊂ R2, and by [BS8] such maps are quasi hyperbolic. One purpose
of the present paper is to extend the work of [BSr] from the context of real, maximal entropy
to the more general setting of quasi hyperbolicity.
Theorem 1 will be proved in §3, and Theorem 2 will be proved in §4.
§1. Invariant families of parametrized curves. [BS8] gives several distinct but equivalent
ways of defining quasi-expansion. One of these is the Proper, Bounded Area Condition, which
makes no explicit reference to expansion. There are also a number of other definitions which
use the pluri-complex Green function G+, which is characterized by the properties:
2
(i) G+ is continuous on C2, G+ = 0 on K+, and G+ > 0 on C2 −K+,
(ii) G+ is pluri-subharmonic on C2, and pluri-harmonic on C2 −K+,
(iii) G+(z) ≤ log(||z||+1)+O(1), and limsup G+(z)/log(||z||+1) = 1.
z→∞
A convenient formula (see [H], [FS], [BS1]) is that G+ is the super-exponential rate of escape
of orbits to infinity: G+(z) = lim (deg(f))−nlog(||fn(z)||+1) .
n→∞
Oneoftheequivalentdefinitionsofquasi-expansionconcernstheexistenceofalargenormal
familyofentirecurvesimbeddedinJ−. Wewillusethisasourdefinitionandderiveanumberof
its properties. We let Ψ be an f-invariant family of injective holomorphic maps ψ : C → J− ⊂
C2 such that ψ′(ζ) 6= 0 for all ζ ∈ C and which satisfy the disjointness (1.1) and normalization
(1.2) conditions:
If ψ ,ψ ∈ Ψ, then either ψ (C) = ψ (C), or ψ (C)∩ψ (C) = ∅ (1.1)
1 2 1 2 1 2
and
maxG+(ψ(ζ)) = 1, (1.2)
|ζ|≤1
We note that if ψ satisfies (1.2), then so does the “rotated” parametrization ψ(eiaζ) for any
a ∈ R. If x = ψ(0), then by the disjointness condition, we may write ψ = ψ . Any other φ ∈ Ψ
x
with φ(0) = x and which also satisfies (1.2) will be rotated reparametrization of ψ . Thus the
x
image of ψ determines the map ψ up to precomposition by a rotation of C.
x x
Let X ⊂ J be a set with J∗ ⊂ X. We let Ψ := {ψ : x ∈ X} be an invariant family as
x
above. Ne give the two principal examples of families Ψ, which exist for all H´enon maps. For
one of them, Ψ , the curves are the unstable manifolds of saddle points and are thus pairwise
S
disjoint. ThisisExample1below, andis whatwas used in[BS8]. ForExample2wework inside
a single slice: all of the curves are the same unstable manifold Wu(q), but the parametrizations
are different.
Example 1. (Saddle Points) We let X := S be the set of periodic points of saddle type.
Suppose that fN(p) = p, and the multipliers of DfN at p are ν ,ν ∈ C with |ν | < 1 < |ν |.
s u s u
For each p ∈ S, there is a uniformization ξ : C → Wu(p) such that ξ (0) = p, and fN(ξ (ζ)) =
p p p
ξ (ν ζ). We may change the parametrization so that ψ (ζ) := ξ (αζ) satisfies (1.2). It follows
p u p p
that Ψ := {ψ : p ∈ S} satisfies the conditions above.
S p
Example 2. (Recentered Unstable Manifold) Let q denote any saddle point, and let
ξ : C → Wu(q) be the uniformization. By [BLS], Wu(q)∩Ws(q) is a dense subset of J∗, and
q
we choose X ⊂ Wu(q)∩J such that J∗ ⊂ X. For y ∈ X, let ζ ∈ C be such that ξ (ζ ) = y.
y q y
We may “re-center” the parametrization of this curve to the point y, i.e., we choose α ∈ C so
that ψ (ζ) := ξ (αζ +ζ ) satisfies (1.2). Thus Ψ := {ψ : y ∈ X} satisfies the conditions
y q y qR y
above.
We say that f is quasi expanding* if Ψ is a normal family, which means that if {ψ } ⊂ Ψ is
j
any sequence, then there is a subsequence {ψ } which converges uniformly on compact subsets
jk
of C to an entire map ψ : C → C2. We let Ψ denote the set of such normal limits, and for
b
x ∈ X we set Ψ := {ψ ∈ Ψ : ψ(0) = x}. There are quasi-expanding maps f for which ψ ∈ Ψ
x
b b b
may fail to be 1-to-1, and there may be ζ where ψ′(ζ ) = 0. Thus the elements of Ψ may not
0 0 x
b
be essentially unique modulo rotation of parameters.
* Ourdefinitionofquasiexpansionisslightly differentfrom[BS8]becauseoftheintroduction
of the set X, which allows us to deal more flexibly with the possibility that J∗ 6= J.
3
Proposition 1.1. If f is quasi-expanding, then Ψ satisfies (1.1) and (1.2).
b
Proof. It is evident that (1.2) must hold. If (1.1) fails, there are ζ ,ζ ∈ C and ψ ,ψ ∈ Ψˆ
1 2 1 2
such that ψ (ζ ) = ψ (ζ ), but ψ (C) 6= ψ (C). Thus x˜ := ψ (ζ ) = ψ (ζ ) is an isolated
1 1 2 2 1 2 1 1 2 2
point of ψ (C)∩ψ (C). On the other hand, ψ is the locally uniform limit of ψ ∈ Ψ (and
1 2 1 1,j
similarly for ψ ). By the continuity of complex intersections, there is an intersection point of
2
ψ (C)∩ψ (C) near x˜ when j is sufficiently large. Now if we choose k sufficiently large, there
1,j 2
is intersection point of ψ (C)∩ψ (C) near x˜. This contradicts (1.1) for the original family
1,j 2,k
Ψ.
For x ∈ X and r > 0 we let B(x,r) ⊂ C2 denote the ball of radius r, centered at x. For
ψ ∈ Ψ , we let D = D (r) denote the connected component of the open set ψ−1(B(x,r))
x ψ ψ
b
containing the origin. By (1.2), ψ is non-constant, and we can choose r > 0 small enough that
ψ : D → B(x,r) is proper, which corresponds to the property that |ψ| = r > 0 on ∂D .
ψ ψ
We may choose a uniform r that works for all x ∈ J∗. By the Maximum Principle, D is
ψ
a topological disk, so by the Riemann Mapping Theorem it is conformally equivalent to the
unit disk. For x ∈ X and ψ ∈ Ψ , the fact that ψ is non constant means that there is an
x
b
integer m ≥ 1 and ~a ∈ C2 such that ψ(ζ) = x +~aζm + ···, where the dots indicate higher
powers of ζ. Thus there are r,ρ > 0 such that dist(ψ(ζ),x) ≥ r for all |ζ| = ρ. This means
that D ⊂ {|ζ| < ρ}. Since |ψ′| is bounded by some number M on |ζ| ≤ ρ, we know that
ψ
{|ζ| < r/M} ⊂ D . By the compactness of Ψ, we have:
ψ
b
Proposition 1.2. For sufficiently small r > 0, there are 0 < ρ < ρ such that {|ζ| < ρ } ⊂
1 2 1
D ⊂ {|ζ| < ρ } for all ψ ∈ Ψ.
ψ 2
b
Let r be as in Proposition 1.2, and let x ∈ X and ψ ∈ Ψ . As in §0, we define Wu :=
bx x,r
ψ(D ), and we denote the family of these disks by Wu := {Wu : x ∈ X}.
ψ r x,r
Proposition 1.3. The set Wu is well defined and does not depend on the choice of ψ ∈ Ψ ,
x bx
and ψ : D → B(x,r) is a proper imbedding. Further, Wu has the following properties:
ψ x
(i) Wu is a nonsingular subvariety of B(x,r).
x,r
(ii) The family Wu has uniformly bounded area: sup Area(Wu ) < ∞.
r x∈X x,r
(iii) X ∋ x 7→ Wu is continuous in the Hausdorff topology.
x,r
Proof. We start by proving uniqueness of the set Wu. By the construction of D given above,
x ψ
it follows that ψ : D → B(x,r) is proper. If there are ψ ,ψ ∈ Ψ with Wu 6= Wu , then
there are points ζ ,ζψ∈ C with 0 < |ζ |,|ζ | < ρ such that1ψ 2(ζ ) =bxψ (ζ ). Nψo1w forψj2= 1,2,
1 2 1 2 2 1 1 2 2
(n) (n)
ψ is the limit of maps ψ , n → ∞. As in the proof of Proposition 1.1, there are points ζ
j j j
(n) (n) (n) (n) (n)
with ζ → ζ as n → ∞, and ψ (ζ ) = ψ (ζ ). This, however, violates (1.1).
j j 1 1 2 2
The proof of item (i) is nontrivial and is a consequence of Proposition 12 of [LP]. Items (ii)
and (iii), on the other hand, are easy consequences of the normality of Ψ. In particular, for (iii)
we use the uniqueness of Wu , together with the fact that ψ : D → B(x,r) is proper. Thus
x,r x ψ
convergence of Wu in Hausdorff topology is a consequence of uniform convergence of ψ .
x,r x
§2. Expanded metric. We continue to work with Ψ as above. If x ∈ X, then f(ψ ) differs
x
from ψ by a linear reparametrization, so there exists λ ∈ C such that
f(x) x
f(ψ (ζ)) = ψ (λ ζ). (2.1)
x f(x) x
4
For λ ∈ C, we define the linear map L (ζ) := λζ. The composition ψ−1 ◦f ◦ψ : C → C
λ f(x) x x f(x)
is an invertible holomorphic map fixing the origin, so by (2.1), we have ψ−1 ◦f ◦φ = L for
f(x) x λx
some λ ∈ C. This means that the action of f on the curves ψ (C) is that the curve through x
x x
is taken to the curve through f(x), and the uniformizing parameters precompose by the linear
maps L : C → C .
λx x f(x)
The condition (1.2) may be interpreted as saying that the unit disk {|ζ| < 1} is the
largest disk centered at the origin and contained in the set {G+ ◦ ψ < 1} ⊂ C . Under f
x x
(equivalently L ) this is taken to the disk {|ζ| < |λ |} ⊂ C . A basic property of G+ is
λx x f(x)
that G+◦f = deg(f)G+, and deg(f) ≥ 2 > 1. Since {|ζ| < |λ |} is then the largest disk inside
x
{G+ < deg(f)}, it follows that |λ | > 1.
x
Now we give a second interpretation of |λ |. For each x ∈ X, we define Eu to be the
x x
1-dimensional subspace of T (C2) spanned by ψ′(0). We let |·| denote the Euclidean norm
x x e
and define a new norm on Eu:
x
|v|
kvk# := e , v ∈ Eu.
x |ψ′(0)| x
x e
With this definition, the (operator) norm of ψ′(0) is 1. By the chain rule, we have
x
D f ·ψ′(0) = ψ′ (0)·λ (2.2)
x x f(x) x
Thus the operator norm of the restriction of D f to Eu with respect to the metric k·k# is
x x x
given by
kD f(v)k#
kD fk# = x f(x) = |λ |, for v ∈ Eu
x x kvk# x x
x
We define three rate of growth functions:
m (r) := maxG+(ψ(ζ)), m(r) := inf m (r), M(r) := sup m (r)
ψ ψ ψ
|ζ|≤r ψ∈Ψ ψ∈Ψ
b b
With this notation, we see that for x ∈ X, |λ | is defined by the condition m (|λ |) = d.
x ψ x
f(x)
Proposition 2.1. If f is quasi-expanding, then the following hold:
(i) M(r) < ∞ for all r < ∞.
(ii) m(r) > 0 for all r > 0
(iii) Let κ be such that M(κ) = deg(f). Then κ > 1, and |λ | ≥ κ for all x ∈ X.
x
Proof. These properties are proved in [BS8] and follow from the normality of the family Ψ.
In fact, each of these conditions is equivalent to quasi expansion (see [BS8]).
Proposition 2.2. If f is quasi-expanding, then for sufficiently large N, f−N maps Wu in-
r
side itself. That is, for each x ∈ X, f−N(Wu ) ⊂ Wu . Further, if y ∈ Wu , then
x,r f−N(x),r x,r
dist(f−n(x),f−n(y)) → 0 like a constant times κ−n as n → +∞.
Proof. Let ρ , ρ are as in Proposition 1.2, let κ is as in Proposition 2.1, and let N be large
1 2
enough that ρ κN > ρ . Thus
1 2
f−NWxu,r ⊂ ψf−N(x)(|ζ| < κ−Nρ2) ⊂ ψf−N(x)(Df−N(x)) = Wfu−N(x),r
For the last assertion, by the normality of Ψ we have |ψ′(ζ)| ≤ C for all ψ ∈ Ψ and |ζ| ≤ ρ .
2
Now for any two points of Vu, we may write them as y′ = ψ (ζ′) and y′′ = ψ (ζ). The distance
x x x
between the ζ-coordinates of f−n(y′) and f−n(y′′) is κ−n|ζ′ − ζ′′|. Thus we conclude that
dist(f−n(y′),f−n(y′′)) ≤ ρ κ−nC, which proves the last assertion.
2
5
Proposition 2.3. If f is quasi-expanding, then for x ∈ J∗ and x′ ∈ Wu ∩J∗ with x 6= x′,
x,r
there is an N > 0 such that fn(x′) ∈/ Wu for n ≥ N.
fn(x),r
Proof. The proof of this is similar to the proof of the last assertion in Proposition 2.2. The
point x is given by ψ (0), and there is ζ′ ∈ D such that x′ = ψ (ζ′). The ζ-coordinate of
x x x
fn(x′) has modulus bounded below by |ζ′|κn, which is larger than ρ for n large, and so it does
2
not belong to D . This means that fn(x′) ∈/ Wu .
fn(x) fn(x′),r
§3. Stable manifolds. We will prove Theorem 1 in this section. We continue to assume
that f is quasi-hyperbolic. We have defined Wu and ξu := ξ for a quasi-expanding map f,
x,r x x
and we will write Ws and ξs for the corresponding local stable manifolds and parametrizing
x x
maps for f−1. When f is hyperbolic, Ws is the family of stable manifolds; and when f is
quasi-hyperbolic but not hyperbolic, Ws still has many of the properties of a family of stable
manifolds for a hyperbolic diffeomorphism.
Proposition 3.1. If f is quasi-contracting, then for all x ∈ J∗, Ws = S f−nWs is a
x n≥0 fn(x),r
manifold consisting of stable points, i.e., Ws ⊂ Ws.
x x
Proof. We let ρ and ρ be as in Proposition 1.2. Thus Ws ⊃ ψs (|ζ| < ρ ). By
1 2 fn(x),r fn(x) 1
quasi-contraction, we have that f−n(Ws ) ⊃ ψs(|ζ| < κnρ ). Thus Ws will contain all of
fn(x),r x 1 x,r
ψs(C). For the second statement, Proposition 2.2, applied to f−1, shows that Ws , and thus
x x,r
Ws is contained in Ws.
x x
We will show that the local disks Ws and Wu have a weak sort of transversality. For
x,r x,r
a set E ⊂ C2 and ǫ > 0, we will let Eǫ denote the ǫ-neighborhood of E, i.e. set of points of
distance less than ǫ from E.
Lemma 3.2. Let f be quasi-hyperbolic, and let r > 0 be the as in §1. Then for r < r < r
1 2
there exist r ,ǫ > 0 with the following property. Let B ⊂ B ⊂ B ⊂ C2 be concentric balls
0 0 1 2
such that B has radius r . Then
j j
(i) For x′,x′′ ∈ B ∩J∗, the intersections Ws ∩B and Wu ∩B are closed subsets of B ,
0 x′,r 2 x′′,r 2 2
(ii) Ws ∩Wu ∩B is nonempty, and
x′,r x′′,r 1
(iii) (Ws )ǫ ∩(Wu )ǫ ⊂ B .
x′,r x′′,r 1
(iv) diam(cid:0){x′,x′′}∪(Wxs′,r ∩Wxu′′,r)(cid:1) → 0 as dist(x′,x′′) → 0.
Proof. First, we know that J∗ ∋ x 7→ Wu is continuous in the Hausdorff topology. Since
x,r
r < r, we may take r > 0 sufficiently small that Wu ∩ B will be closed in B . Thus
2 0 x r2 r2
Wu ∩B and Ws ∩B are subvarieties of B . Since these varieties do not actually coincide
x,r r2 x,r r2 r2
in a neighborhood of x, we may choose r small enough that Wu ∩Ws ∩B = {x}. The
2 x,r x,r r2
intersection of complex varieties has a continuity property. Specifically, for any r > 0, we
1
may choose r > 0 sufficiently small that the intersection Wu ∩Ws ∩B is contained in B .
0 x′ x′′ r2 r1
Further, we may choose ǫ > 0 sufficiently small that the ǫ-neighborhood is also contained in
B . Finally, we note that (iii) implies that the intersection points of Ws ∩ Wu cannot
r1 x′,r x′′,r
escape to the boundary of B , so (iv) is a consequence of continuity in the Hausdorff topology.
1
If we set g := f|J∗, then we see that the manifolds Ws give the stable sets for g in the
following sense:
6
Theorem 3.3. If f is quasi-hyperbolic, then for all x ∈ J∗, Ws ∩J∗ = Ws ∩J∗.
x x
Proof. By Proposition 3.1, we need to show the containment ‘⊃’. Suppose that z ∈ Ws ∩J∗.
x
We will show that for n sufficiently large, the local varieties Ws and Ws intersect.
fn(z),r fn(x),r
Thus by Proposition 3.1, we have z ∈ Ws.
x
Now let r ,r ,r and ǫ be as in Lemma 3.2. There exists N sufficiently large that fn(z) ∈
0 1 2
B(fn(x),r ) for all n ≥ N. We may suppose that Ws and Ws are disjoint, for
0 fn(x),r fn(z),r
otherwise we would have Ws = Ws , and thus Ws = Ws. By Lemma 3.2, the set
fn(x) fn(z) x z
S := Wu ∩Ws(fn(z),r) is nonempty for all n ≥ N. On the other hand, fj(S ) = S ,
n fn(x),r n n+j
which contradicts Proposition 2.3.
We also have a local uniqueness result for local varieties.
Corollary 3.4. Let f bequasi-hyperbolic, and let x ∈ J∗. If U is a sufficiently small neighbor-
hood of x, and V is an irreducible subvariety of U with V ∩U ⊂ Ws, then Ws ∩U = V ∩U.
x x x x,r x
Proof. By Theorem 3.3, we know that J∗ ∩V = J∗ ∩Ws . Since J∗ is an infinite set with
x x,r
accumulation points, the result follows.
We conclude this section with a result that is proved much along the lines of Theorem 9.6
of [BLS]:
Theorem 3.5. Let f be quasi-hyperbolic, and let T ,T ⊂ J∗ be closed, invariant subsets.
1 2
Then S Ws ∩J∗ and (S Ws)∩(S Wu) are dense subsets of J∗.
x∈T1 x x∈T1 x y∈T2 y
§4. Characterization of hyperbolicity. This section is devoted to a proof of Theorem 2.*
We start by recalling some notation and results from [BS8]. For ψ : C → C2, we define the
order at ζ = 0 to be Ord(ψ) := min{m ≥ 1 : ψ(m)(0) 6= 0}. For x ∈ J∗, we define τs/u(x) =
max Ord(ψ). The quantity τu(x) measures the failure of Wu to be a lamination in a
ψ∈Ψˆs/u
x
neighborhood. More precisely, let π : C2 → Eu denote a linear projection. Then for y ∈ J∗
x
close to x, π : Wu → Eu is a branched cover over a neighborhood of x. By local folding, we
y,r x
mean that there will be y near x for which the local branching order is τu(x). Thus there is no
neighborhood of x on which Wu can be a lamination. Looking at Ws at the same time, we see
that there will be y′,y′′ ∈ J∗ such that Ws ∩Wu contains τs(y′)τu(y′′) points near x.
y′,r y′′,r
Consider ψ ∈ Ψu with j = Ord(ψ ). We have ψ (ζ) = x+a ζj +···, where a ∈ C2 is
x x x x j j
nonzero, and ‘···’ indicates terms of higher order. Further, a = ψ(j)(0)/j! spans the unstable
j
tangent space Eu. Let us define
x
(cid:12)ψ(j)(0)(cid:12)
γu(x) := inf (cid:12) (cid:12)
ψx∈Ψˆux,Ord(ψx)=τu(x)=j(cid:12)(cid:12) j! (cid:12)(cid:12)e
where |·| denotes the euclidean metric on C2. For x ∈ J∗ with τu(x) = j, we define a metric
e
on a vector v ∈ Eu by
x
|v|
kvk#,j := e
x γu(x)
* It might appear that we could obtain a result stronger than is stated in Theorem 2, since
the paper [F] claims that: If J∗ is a hyperbolic set, then J = J∗. However, the proof given in
[F] seems to be incomplete.
7
This metric is uniformly expanded on the set {x ∈ J∗ : τu(x) = j}, but it need not be bounded
with respect to |·| as x approaches points x′ ∈ J∗ where τu(x′) > j. We define
e
J∗ := {x ∈ J∗ : τs(x) = ms,τu(x) = mu}.
ms,mu
We say thatthepair(ms,mu) ismaximal if J∗ 6= ∅, andwheneverms ≥ ms andmu ≥ mu,
ms,mu 1 1
then J∗ = ∅ unless ms = ms and mu = mu. If (ms,mu) is maximal, the metrics k·k#,ms,
ms,mu 1 1
1 1
and k · k#,mu are equivalent to | · | on Es and Eu on J . It follows that: J∗ is a
e x x ms,mu ms,mu
compact, hyperbolic set whenever (ms,mu) is a maximal pair. From [BS8] we have:
Theorem 4.1. J∗ is a dense, open subset of J∗. Further, f is uniformly hyperbolic on J∗ if
1,1
and only if J∗ = J∗.
1,1
Proof of Theorem 2. If J∗ = J∗, then f is uniformly hyperbolic, and thus the stable and
1,1
unstable families Ws and Wu are transverse. Conversely, suppose that J∗ is a proper subset
1,1
of J∗. In this case, there is a maximal pair (ms,mu) 6= (1,1), and J∗ := J∗ is a hyperbolic
0 ms,mu
subset of J∗. We may suppose that ms > 1. Given δ,ρ > 0, we cover J∗ with finitely many
0
compact sets X such that for each ι there is a translation and rotation of coordinates so that
ι
s/u
X ⊂ ∆ ×∆ . WesetN = ∆ ×∆ . Forx ∈ X , weletW denotetheconnectedcomponent
ι δ δ ι ρ ρ ι x,ρ
s/u
of W ∩N containing x. Shrinking δ,ρ if necessary, we may assume that for x ∈ X , each
x,ρ ι ι
Wu is “horizontal” in N , i.e., a graph over the first coordinate, and each Ws is “vertical”.
x,ρ ι x,ρ
Set Ws,u = {Ws,u : x ∈ X }. By the hyperbolicity of J∗, there are cone fields with respect to
Xι x,r ι 0
which f is uniformly expanding in the horizontal direction of N and uniformly contracting in
ι
the vertical direction.
Now let us choose a neighborhood N0 of J0∗ such that f(N0) ⊂ SιNι, and τs < ms on
Wu −N . This is possible because J∗ ∩Wu is dense in J∗ by Corollary 3.5. If τs = ms on
J∗ 0 J∗
0 0
J∗ ∩Wu , then by the upper-semicontinuity of τs, it would follow that τs = ms on all of J∗,
J∗
0
which is not possible by Theorem 4.1.
Figure 1.
For y ∈ J∗ there is a sequence (y ) ⊂ J∗ such that y → y , and ψs ∈ Ψs converges
∞ 0 j j ∞ yj yj
to a map ψs with the property that Ord(ψs ) = ms. We may suppose that y ∈ X , so
∞ ∞ ∞ ι
that Ws is horizontal in N . In order for ψs to have order ms > 1, it is necessary that
y∞,ρ ι ∞
the nearby Ws are ms-fold branched covers with respect to the coordinate projection to the
yj,ρ
tangent space of Ws . Since Ws is a lamination, we see that we cannot have y ∈ Ws .
y∞,ρ Xι j Xι
Since y ∈ J∗, Wu is horizontal in N = ∆ ×∆ in the sense that it is a graph over the first
∞ 0 y∞ ι ρ ρ
factor ∆ and bounded away from ∆ ×∂∆ρ. It follows that the intersection Ws ∩Wu
ρ ρ yj,r y∞,r
8
consists of ms points. By Proposition 2.2, these points get closer together as we map forward
under f.
Now since y ∈/ Ws , it follows that fn(y ) cannot remain in N ∩N for all n > 0. Thus
j Xι j ι 0
there is an nj > 0 such that fk(yj) ∈ N0 for 0 ≤ k < nj, and fnj(yj) ∈ Nι −N0. We may
pass to a subsequence so that the points z := fnj(y ) all belong to the same set N −N . By
j j ι 0
hyperbolicity, Wu is a family of graphs horizontal in N , and we conclude that a subsequence
Xi ι
of the points z converges to a point z ∈ Wu .
j ∞ Xi
Now we consider themultiplicity of the intersection between Ws and Wu at z . The
z∞,ρ z∞,ρ ∞
set Ws ∩Wu consists of ms points, and by part (iv) of Lemma 3.2, these points coalesce
yj,r y∞,r
as j → ∞. Thus by the continuity of the intersection multiplicity in the complex setting, the
intersection multiplicity is ms. On the other hand, let ψˆs be the limit of the maps ψs . It
∞ zj
follows thatOrd(ψˆs ) ≤ τs(z ) < ms. We concludethatthecurves mustintersect tangentially.
∞ ∞
Proposition 4.2. If a point x ∈ J∗ and Wu(x) and Ws(x) are tangent to order k then
ms,mu
the forward limit set of x is contained in J∗ with p ≥ ms and q ≥ (k +1)mu the backward
p,q
limit set of x is contained in J∗ with p ≥ (k+1)ms and q ≥ mu.
p′,q′
Proof. We prove the first statement. The second statement follows by considering f−1.
Consider the case when ms = mu = 1. Say that ψ (ζ) parametrizes Wu(x) and φ (ζ)
x x
parametrizes Ws(x). Since Wu(x) and Ws(x) are tangent to order k we can assume that
for some c 6= 0, ψ (ζ) = φ (cζ) + ··· where the dots represent terms or order greater than
x x
k. Write fn(ψ (ζ)) = ψ (λ ζ) and fn(φ (ζ)) = φ (µ ζ). For some κ > 1 we have
x fn(x) n x fn(x) n
λ ≥ κn and µ ≤ κ−n.
n n
Now fn(ψ (ζ)) = fn(φ (cζ)) up to terms of order greater than k. Write ψ (ζ) =
x x fn(x)
fn(x)+~a ζ+~a ζ2+···andφ (ζ) = fn(x)+~b ζ+~b ζ2+···. Thenwehaveψ (λ ζ)
1,n 2,n fn(x) 1,n 2,n fn(x) n
is equal to φ (cµ ζ) up to order k so by comparing coefficients we get ~a λj =~b cjµj
fn(x) n j,n n j,n n
for j = 1,...,k. By normality |b | is bounded by a constant depending on j but not n so
j,n
|~a | ≤ C κ−2n|cj|. We conclude that for j = 1,...,k, |~a | → 0 as n → ∞. In particular if
j,n j j,n
x′ is in the forward limit set of x then taking φx′ to be a convergent subsequence of φfn(x) the
coefficients of φx′(ζ) vanish for j = 1,...,k.
If there is a map ϕ in Φu which vanishes to order ℓ then ϕ (ζ) can be written as the
x x x
composition of a map ξ and a map α of order ℓ. Arguing as above we see that the limits
x
ϕ α vanish to order at least (k+1)ℓ.
fn(x)
Corollary 4.3. For f quasi-hyperbolic there is a bound on the order of tangency between
stable and unstable manifolds.
Proof. In [BS8] it is shown that the set J∗ is empty for ms and mu large.
ms,mu
Theorem 4.4. If f is quasi-hyperbolic, then the condition that J+ is laminated in a neigh-
borhood of J∗ is equivalent to the condition that J− is laminated a neighborhood of J∗; and
either condition is equivalent to hyperbolicity on J∗.
Proof. If f is quasi-hyperbolic but not hyperbolic, then by Theorem 2 there is a tangency of
order k ≥ 1 so by Proposition 4.2 both J∗ and J∗ are non-empty so neither J+ nor J− are
1,k k,1
laminated in a neighborhood of J∗.
In [RT] an example is given of a H´enon diffeomorphism for which J+ is laminated. On the
other hand, this example has a semi-parabolic fixed point, which implies that it is not quasi
expanding, and thus we conclude from the results of [BS8] that J− is not laminated.
9
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arXiv:1411.3824
Eric Bedford
Stony Brook University
Stony Brook, NY 11794
[email protected]
John Smillie
Mathematics Institute
Zeeman Building
University of Warwick
Coventry CV4 7AL
[email protected]
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