Dynamical Objects for Cohomologically Expanding Maps
Dynamical Objects for Cohomologically
Expanding Maps.
John W. Robertson
November 4, 2018
John W. Robertson1
Wichita State University
Wichita, Kansas 67260-0033
Phone: 316-978-3979
Fax: 316-978-3748
robertson@math.wichita.edu
Abstract
The goal of this paper is to construct invariant dynamical objects
for a (not necessarily invertible) smooth self map of a compact mani-
fold. We prove a result that takes advantage of differences in rates of
expansion in the terms of a sheaf cohomological long exact sequence
to create unique lifts of finite dimensional invariant subspaces of one
term of the sequence to invariant subspaces of the preceding term.
This allows us to take invariant cohomological classes and under the
right circumstances construct unique currents of a given type, includ-
ing unique measures of a given type, that represent those classes and
are invariant under pullback. A dynamically interesting self map may
have a plethora of invariant measures, so the uniquess of the con-
structed currents is important. It means that if local growth is not
too big compared to the growth rate of the cohomological class then
the expanding cohomological class gives sufficient “marching orders”
to the system to prohibit the formation of any other such invariant
1Research partially supported by a Wichita State University ARCS Grant.
current of the same type (say from some local dynamical subsystem).
Because we use subsheaves of the sheaf of currents we give conditions
under which a subsheaf will have the same cohomology as the sheaf
containing it. Using a smoothing argument this allows us to show
that the sheaf cohomology of the currents under consideration can be
canonically identified with the deRham cohomology groups. Our main
theorem can be applied in both the smooth and holomorphic setting.
MSC: 37C05, 32H50, 18F20, 55N30
1 Introduction
Our purpose is to construct invariant dynamical objects for a self map f : X →
X of a compact topological space. We make use of sheaf cohomology and
differences in rates of expansion in different terms of a long exact sequence
to construct invariant sections of a sheaf. We will show that there are in-
variant degree 1 currents (or eigencurrents) corresponding to each expanding
eigenvector of H1(X,R). We also show that successive preimages of suffi-
ciently regular degree one currents converge to one of these eigencurrents.
We show that if most of the expansion f : X → X is ”along” an invariant
cohomological class v ∈ Hk(X,R) then there is an invariant current c in that
cohomology class and other sufficiently regular currents in the same class
converge to c under successive pullback.
The group cohomology of Z acting on a space of functions on X via pull-
back has been studied in the context of dynamical systems [Kat03]. This
work seems related to ours, but to be pursued in an essentially different di-
rection. Our map f is not assumed to be invertible, so there is not necessarily
a Z action, only an N action. Also, we use sheaves rather than functions and
make substantial use of cohomological tools. Most importantly, we are par-
ticularly interested in the construction of invariant currents, especially when
the current is some sense unique.
Our results are actually motivated by results in higher dimensional holo-
morphic dynamics showing the existence of a unique closed positive (1, 1) cur-
rent under a variety of circumstances (just about any recent paper on higher
dimensional holomorphic dynamics either proves such results or makes essen-
tial use of such results, see e.g. [FS92], [HOV94], [HOV95], [BS91a], [BS91b],
[BS92], [BLS93], [BS98a], [BS98b], [BS99], [Can01], [McM02], [FS94a], [FS94b],
[FS95b], [FS01], [FS95a], [JW00], [FJ03], [Ued94], [Ued98], [Ued97], and
[DS05]).
While invariant measures have been a focal point in dynamics, it seems
that invariant currents also have an imporant role to play. We will show under
mild conditions that if some degree one cohomological class of a smooth self
map f of a compact manifold is invariant and expanded there is necessarily
a invariant degree one current of a certain type representing that class. We
obtain analogous results for higher degree currents given bounds on the local
growth rates of f . The uniqueness of these classes is significant. It seems
clear that one could modify a map locally near a fixed point to obtain other
invariant currents of the same type without affecting the topology. Thus
our results also say that any local modification that created an invariant
current of the given type must violate the local growth conditions. In other
words, as long as things do not grow too fast compared to the growth rate of
the cohomology class, the expansion of the cohomology class gives sufficient
“marching orders” to points that no other invariant cohomological class of the
given type can be created by purely local dynamical behavior. Our results
give explicit conditions under which uniqueness is guaranteed. For degree
one currents, no restriction on local growth rates is necessary for our results.
2 Cohomomorphisms
We will make use of sheaves in this paper. There are two standard def-
initions of sheaves on a topological space X, one as a topological space
([Bre97],[GR84]), and one as a functor on the category TopX satisfying var-
ious axioms ([Har77],[Wei97]). Since we will often want to make use of a
topology on sections of a sheaf A that differs from the topology these inherit
using the topological definition of a sheaf, we will instead use the functor
definition of a sheaf.
Our sheaves will always be sheaves of K modules over some fixed field K.
We will require that K have an absolute value for which K is complete.
Given a continuous map f : X → Y and sheaves A and B on X and
Y respectively, an f -cohomomorphism is a generalized notion of a pullback
from B to A through f . Different types of geometric objects pull back
differently, and this allows us to handle all cases at once.
We take the following facts from from [Bre97] page 14–15.
Definition 1. If A and B are sheaves onX and Y then an “f-cohomomorphism”
k : B → A is a collection of homomorphisms kU : B(U) → A (f−1(U)), for
U open in Y , compatible with restrictions.
Note that if A is a sheaf on X and f : X → Y is continuous then there
is a canonical cohomomorphism f∗A ; A where f∗A is the direct image of
A , i.e. given an open U ⊂ Y , f∗A (U) = A (f−1(U)).
Remark. Given a continuous map f : X → Y of topological spaces X and
Y and sheaves A and B on X and Y respectively, all f -cohomomorphisms
f : B ; A are given by a composition of the form
j→ f∗A
f∗→ A
where j : B → f∗A is a sheaf homomorphism, and each such composition is
seen to given an f -cohomomorphism.
The usual notion of “a morphism of sheaves on X” is the same as an idX
cohomomorphism of sheaves on X.
2.1 Cohomomorphisms and Γ.
The functor Γ returns the global sections of that sheaf. Given a morphism
φ : A → A ′ of sheaves on X, Γφ is just the homomorphism A (X)→ A ′(X).
Given sheaves A and B on X and Y and given f : X → Y continuous
then for a sheaf cohomomorphism F : B → A one defines ΓF to be the
homomorphism B(Y ) → A (X). This extends Γ to be a functor on the
category of topological spaces with an associated sheaf where morphisms are
given by cohomomorphisms.
3 Invariant Global Sections
Fix a continuous self map f : X → X of a topological space X. We will
be interested in f self cohomomorphisms of sheaves A on X. As we will
typically have several sheaves of interest on X, each with a corresponding
f self cohomomorphism, we let fA : A ; A be the default notation for an
f -cohomomorphism of A .
Assume that X is a manifold and that
p→ B q→ C
is a short exact sequence of sheaves on X. Let f : X → X be a continuous
self map of X and assume further that we are given f self cohomomorphisms
of each of these sheaves and that
// B q
commutes. We will say that a commutative diagram as in (1) is an f self-
cohomomorphism of the sequence A → B → C .
Applying the functor Γ to this diagram, the rows can be extended in the
usual long exact sequence. The resulting diagram is commutative ([Bre97]
page 62).
0 // A (X)
C (X)
H1(X,A )
· · ·
0 // A (X)
// B(X)
// C (X)
// H1(X,A )
// · · ·
One can think of B as providing local potentials for members of C and
of A as being those potentials which give rise to the zero member of C .
It will be assumed that the reader is familiar with interpreting H1(X,A )
as classifying equivalence classes of bundles with transition functions in A .
We will frequently refer to members of H1(X,A ) as bundles. Sections of
such bundles will be assumed to be given locally by local sections of B,
so that every member c of Γ(C ) is given locally by potentials in B, and
these potentials, taken together, are a section of the corresponding bundle
δ(c) ∈ H1(X,A ).
Convention 1. We will frequently refer to a member v of H1(X,A ) as a
bundle, to a member c ∈ Γ(C ) as a divisor and if δ(c) = v we will call c a
divisor of the bundle v. We think this substantially adds to the readability of
the paper.
Definition 2. The support of a divisor c ∈ Γ(C ) is defined to be the com-
plement of the union of all open sets U such that c
Lemma 3. If an open set U lies outside the support of some c ∈ Γ(C ) then
f−1(U) lies outside the support of fC (c)
Proof. We note that by the definition of an f -cohomomorphism fC : C → C ,
since the cohomomorphism fC on C (U) is a homomorphism from C (U) to
C (f−1(U)) and the induced action of fC on Γ(C ) restricted to U must agree
with its action C (U) → C (f−1(U)), then if an open set U is outside the
support of c then f−1(U) is outside the support of of fC (c).
The following conditions for a given v ∈ H1(X,A ) will be of interest:
Definition 4. We will refer to a bundle v ∈ H1(X,A ) for which (H1p)(v) =
0 as being closed.
Note that this notion depends upon the exact sequence A → B → C ,
and not just on v. If B is γ acyclic then every member of H1(X,A ) is closed.
Definition 5. We will call a bundle v ∈ H1(X,A ) base point free if for every
x ∈ X there is some divisor c ∈ Γ(C ) associated to v whose support does
not contain x.
Lemma 6. If B is soft, X is a regular topological space, and a ∈ H1(X,A )
is a closed bundle then a is base point free.
Proof. From the long exact sequence there is some c′ ∈ Γ(C ) with δ(c′) = a
and given any point x ∈ X, from the fact that B
� C the germ c′x of c
at x is the image under qx of some germ b
x of Γ(B) at x. Choose an open
neighborhood U of x on which there is some b′ ∈ B(U) with b′x = b′′x. The
topological assumption on X implies that there is a neighborhood V b U
of x. The fact that B is soft implies there is some b ∈ Γ(B) such that
. Then c = c′ − b ∈ Γ(C ) has δ(c) = a and x 6∈ Supp(c).
Definition 7. We will refer to a bundle a ∈ H1(X,A ) such that fA (a) = λ·a
for some λ ∈ C as a λ eigenbundle.
We also find it useful to introduce a relevant notion of expansiveness
of a map f : X → X relative to a base point free closed eigenbundle v ∈
H1(X,A ).
Definition 8. Given a base point free closed eigenbundle v ∈ H1(X,A )
then we say that f is cohomologically expansive at x for v if for any open
neighborhood U of x and any divisor c ∈ Γ(C ) of v, the set U intersects the
support of fkC (c) for all sufficiently large k.
Remark. It is a corollary of the definition that the set of points at which
f is cohomologically expansive for v is closed and forward invariant. If
Supp fkC (c) = f
−k(Supp(c)) for each c ∈ Γ(C ) then the set of cohomolog-
ically expansive points is totally invariant.
The notion of being cohomologically expansive at x for v means roughly
that under iteration by f small neighborhoods U of x always grow to cover
enough of X that the pullback of the bundle v to the set fk(U) is a nontrivial
bundle on fk(U) whenever k is large.
We show that if B is soft and X is a compact metric space then some
minimal expansion takes place at points where f is cohomologically expansive
for a closed eigenbundle a ∈ H1(X,A ).
We use B�(x) to denote the ball of radius � about x.
Lemma 9. Let X be a compact metric space. If B is soft and v is a closed
eigenbundle then there exists δ > 0 such that for every � > 0 there exists
some K > 0 such that if f is cohomologically expanding at x then for every
k > K, diam fk(B�(x)) > δ.
Proof. The bundle v is base point free by Lemma 6. Using compactness we
can conclude that there is a finite open cover U1, . . . , U` of X such that for
each j, Uj is disjoint from Supp cj for some cj ∈ Γ(C ) with δ(cj) = v. We
will prove the lemma by contradiction. Let δ be the Lebesgue number of the
cover U1, . . . , U`. If the lemma is false there is some � > 0 and some increasing
sequence kn and points xn at which f is cohomologically expansive such that
diam fkn(B�(xn)) ≤ δ for each n. By going to a subsequence if necessary we
can assume xn converges to a point x∞. Letting U = B 1
�(x∞) we see that
U ⊂ B�(xn) for all large n and thus there is some one cj of c1, . . . , c` such that
fkn(U) is disjoint from Supp cj for infinitely many values of n. Consequently
U is disjoint from Supp fknC (cj) for infinitely many n, contrary to x∞ being
a point at which f is cohomologically expansive for v.
We included Lemma 9 to show that our notion of cohomological expansion
is genuinely expansive. However, depending on the nature of A , being coho-
mologically expansive can imply that neighborhoods grow a great deal under
iteration indeed. In Lemma 10 we show that given any closed set K such that
the pullback of a fixed point free closed eigenbundle a ∈ H1(X,A ) to K is a
trivial bundle then any neighborhood U of a point at which f is cohomolog-
ically expanding for a is so expanded under iteration that fk(U) 6⊂ intK for
all sufficiently large k. The collection of such sets K typically contains very
large sets about every point so no matter where fk(x) is the conclusion that
fk(U) does not lie in any intK implies some points of fk(U) must lie far away
from fk(x). The point is roughly that large iterates of any neighborhood of
x can not be homotopically contracted to a point in X.
Lemma 10. If B is soft, then for any closed set K ⊂ X such that the image
of H1(X,A ) → H1(K,A
) is zero, given any divisor c ∈ Γ(C ), there is
another divisor c′ ∈ Γ(C ) associated to the same bundle and c′ is supported
outside the interior of K. Consequently, if f is cohomologically expansive
at x ∈ X for some base point free closed eigenbundle a ∈ H1(X,A ) then
necessarily for any neighborhood U of x, fk(U) 6⊂ intK for all large k, where
intK is the interior of K.
Proof. We use the commutative diagram
H0(X,B)
Γq //
H0(X,C )
H1(X,A )
H0(K,B
Γq // H0(K,C
δ // 0
which we have written using H0 instead of Γ so it is clear what the ambi-
ent space is in each case. From exactness there exists some β ∈ H0(K,B
such that δ(β) = c
. Then since B is soft the mapH0(X,B)→ H0(K,B
is surjective so there is some b ∈ Γ(B) = H0(X,B) such that b
= β. Then
c′ = c − (Γq)(b) has δ(c′) = δ(c) and c′
= 0 so Supp(c′) is disjoint from
the interior of K.
It is easy to see that if f is cohomologically expansive at x ∈ X for some
fixed point free closed eigenbundle a ∈ H1(X,A ) then necessarily for any
neighborhood U of x, fk(U)∩Supp c 6= ∅ for all large k for any c ∈ Γ(C ) such
that δ(c) = a. Hence fk(U) can not lie in the interior of K for any large
Convention 2. We let K be either R or C, although our central theorems
only require K to be a complete field with an absolute value.
The following Theorem takes advantage of the fact that in an exact se-
quence the eigenvalues of members of nonadjacent members of the sequence
do not have to agree to give conditions under which one can uniquely “lift”
fixed members of one term of the exact sequence to a fixed member of the pre-
ceding term. Interpreted as a statement in the context of sheaf cohomology
we will be able to use this Theorem to make dynamical conclusions.
The theorem shows that each closed eigenbundle of the induced map
fA : H
1(X,A ) → H1(X,A ) with sufficiently large eigenvalue has a unique
associated invariant divisor c ∈ Γ(C ).
Definition 11. Given any finite dimensional K vector space V along with
a linear map g : V → V and any positive real number r, we let the r chron-
ically expanding subspace of V be the span of the subspaces associated2 to
eigenvalues of absolute value greater than r. We refer to the 1 chronically
expanding subspace simply as the chronically expanding subspace.
Theorem 12 (Unique Invariant Subspace Theorem). We will assume the
following:
• f : X → X is a continuous self map of a topological space X.
• We are given an f self cohomomorphism of a short exact sequence of
sheaves on X,
p→ B q→ C
• Γ(B) is a Banach space over K, and there exists some α, d ∈ R>0 such
that ‖ΓfBk(B)‖ ≤ d · αk‖B‖ for k ∈ N, B ∈ Γ(B),
• Γ(C ) is a topological vector space over K.
• If a sequence Ci ∈ Γ(C ) of divisors converges to another divisor C∞
then the support of C∞ is contained in the closure of the union of the
supports of Ci.
• The maps ΓfC and Γq are continuous.
• We are given a finite dimensional H1(fA ) invariant subspace W of the
α chronically expanding subspace of H1(X,A ). We also require W to
be comprised only of closed bundles.
2Meaning for each eigenvector λ we include not just the λ eigenspace, but also every
v ∈ V such that (g − λ · idV )n(v) = 0 for some positive integer n.
Then given any K linear map s : W → Γ(C ) such that δs = idW there is a
K linear map τ : W → Γ(B) satisfying
κ := lim
(ΓfC )
ksgk = s+ (Γq)τ (3)
where g : W → W is the inverse of H1fA
. Under iterated pullback the
rescaled pullbacks of any divisor C ∈ Γ(C ) of a bundle w ∈ W converge
toward the invariant plane of divisors κ(W ) ⊂ Γ(C ). The map κ : W →
Γ(C ) is the unique map making the diagram
wwo o
Γ(C )
// H1(X,A )
wwo o
// Γ(C )
// H1(X,A )
commute. Finally, for any basepoint free eigenbundle v ∈ W the support of
the corresponding invariant divisor κ(v) ∈ Γ(C ) is contained in the set of
points on which f is cohomologically expansive for v.
Proof. We note that δ
(ΓfC )sg−s
= 0 and so there is a map σ : W → Γ(B)
such that (Γq)σ = (ΓfC )sg − s.
Define Φ: Hom(W,Γ(B)) → Hom(W,Γ(B)) by Φ(σ) = (ΓfB)σg−1. We
will show that the sequence of maps Φk is exponentially contracting on
Hom(W,Γ(B)). Fix a norm ‖ · ‖ on W . The assumption that W lies in
the α chronically expanding subspace of H1(X,A ) implies that there exists
some β > α and some c > 0 such that ‖g−k(w)‖ ≤ cβ−k‖w‖ for k ∈ N,
w ∈ W . This with the assumption on the rate of expansion of ΓfB easily
implies that
‖Φk(φ)(w)‖ = ‖(ΓfB)k(φ(g−k(w)))‖ ≤ cd
‖φ‖ · ‖w‖
Thus Φk is an operator of norm no more than cd
, where α < β.
Letting τk = σ + Φ(σ) + Φ
2(σ) + · · · + Φk(σ) then limk→∞ τk converges
to some map τ . It is easily confirmed that (Γq)τk = (ΓfC )
ksg−k − s. Equa-
tion (3) then follows by continuity of Γq.
The conclusions about the map κ are easy consequences of its definition.
For the final conclusion note that if we just let W be the span of v then
we have already shown that if C is the unique invariant member of Γ(C )
associated to v then for any divisor c′ ∈ Γ(C ) satisfying δ(c′) = v letting
λ be the eigenvalue of v we can write c′ = κ(v) + (Γq)(b) and equation 3
becomes (ΓfC )
kc′/λk = κ(v) + (Γq)(ΓfB)
kbλk where the final term goes to
zero as k →∞ (by our assumptions on growth rates of g−1 and ΓfB). Hence
(ΓfC )
k(c′)/λk converges to c = κ(v). If U is any open subset of X and if the
support of c′ is disjoint from fn(U) for arbitrarily large values of n, then the
support of (ΓfC )
n(c′) must be disjoint from U for arbitrarily large values of
n. Since, rescaled, these converge to c then U must lie outside the support
of c.
Remark. While we have not formally required X to be compact, the re-
quirement that Γ(B) be a Banach space makes this the main case in which
Theorem 12 is apt to have interesting applications.
Theorem 12 shows that among all members of Γ(C ) representing a coho-
mology class in W there is a unique invariant linear subspace which can be
identified with W and all other such members of Γ(C ) are contracted to this
invariant copy of W in Γ(C ) under (rescaled) pullback.
Corollary 13. Assume that the hypothesis of Theorem 12 are satisfied, and
that g : W → W is dominated by a single simple real eigenvalue r > 0 with
eigenvector v. Let C ≡ κ(v) be the unique invariant divisor of v. Then given
a divisor C′ ∈ Γ(C ) of any w ∈ W the successive rescaled pullbacks fkC (C
′)/rk
converge to a multiple (possibly zero) of C.
Proof. This is a direct consequence of equation (3).
The assumption that g : W → W is dominated by a single simple real
eigenvalue is meant to handle the most typical situation, and is not an es-
sential restriction.
Remark. Given that for a fixed f : X → X the category of SC sheaves A
on X endowed with an f self cohomomorphism F is an abelian category
with enough injectives, then the functor Fixed Γ which gives the fixed global
sections of A under F will be left exact and its right derived functors should
be of dynamical interest. In the case where A is a sheaf of functions and
f is invertible this is just group cohomology with the group Z acting on
Γ(A ) and has been an object of study for some time (see, e.g. [Kat03]). We
anticipate studying the case of more general sheaves A and the right derived
functors of the composition Fixed Γ in a future paper, including the case of
endomorphisms.
3.1 Regularity and Positivity
Typically our regularity results for the members invariant plane κ(W ) will
be most easily described in terms of B rather than C . We therefore make
the following definition.
Definition 14. Given a subsheaf B′ ⊂ B we will say a divisor C ∈ Γ(C ) has
local B′ potentials if C ∈ Γ(q(B′)). This is equivalent to requiring that about
each point x ∈ X there is an open neighborhood U and some B′ ∈ B′(U)
such that q(B′) = C
The proof of Theorem 12 implicitly provides a method to prove regularity
results for members of the invariant plane κ(W ). We make this explicit as a
corollary (of the proof).
Corollary 15. Assume we are given f : X → X and a short exact sequence
of sheaves A
p→ B q→ C satisfying the hypothesis of Theorem 12. Assume
that B′ is a subsheaf of B and that ΓfB(B
′) ⊂ B′. Let C ′ be the image of
B′ under q : B → C . Let A ′ ⊂ A be the kernel of q : B → C ′. Assume
that the canonical map H1(X,A ′) → H1(X,A ) is injective. Assume that
there are basis members w1, . . . , wk of W with divisors each of which has local
potentials in B′. Let r be the the inverse of the absolute value of the largest
eigenvalue of g−1 (so for all j ≥ 0, g−j is an operator of norm no more
than cr−j for some c > 0) Finally assume that for any sequence of numbers
aj, j = 0, 1, 2, . . . such that |aj| is no more than a constant times r−j as
j →∞ then for B ∈ Γ(B′) the exponentially decaying sequence
a0 B + a1 (ΓfB)(B) + a2 (ΓfB)
2(B) + · · · (4)
converges in the Banach space structure on Γ(B) to a member of Γ(B′).
Then the map κ : W → Γ(C ) lands in Γ(C ′).
Proof. Since W lies in the α chronically expanding subspace of W then neces-
sarily α/r < 1. Thus the terms of equation (4) have exponentially decreasing
norms and the series is exponentially decaying.
By the assumption of a divisor in Γ(C ′) for each member wj of a basis
then the map s : W → Γ(C ) in Theorem 12 can be assumed to land in Γ(C ′).
Then (ΓfC )sg
−1− s lands in Γ(C ′) and satisfies δ((ΓfC )sg−1− s) = 0. Since
H1(X,A ′) → H1(X,A ) injects it easily follows that for each wj one can
choose σ(wj) to be a member Bj of Γ(B)
′. Using the basis w1, . . . , wk to write
g−1 as a matrix A, and letting aij,` be the ij entry of A
` (so for each ij, aij,` is
bounded by a constant times r−`) we see that τ`(wj) = Bj + (ΓfB)(a1j,1B1 +
· · ·+ akj,1Bk) + (ΓfB)2(a1j,2B2 + · · ·+ akj,2Bk) + · · ·+ (ΓfB)`(a1j,`B1 + · · ·+
akj,`Bk). Gathering all the B1 terms, B2 terms, etc... from the right hand
side we see that τ = limk→∞ τk is a member of Γ(B
′) and thus that κ lands
in Γ(C ′) by equation (3).
The following trivial observation will suffice for our needed positivity
conclusions.
Observation. Assume we have an f self cohomomorphism of a short exact
sequence of sheaves A
p→ B q→ C satisfying the hypothesis of Theorem 12,
and also a subsheaf C ′ ⊂ C such that
1. C ′ is closed under multiplication by R>0. Note that C ′ is not necessarily
a sheaf of K modules, or even of groups.
2. fC (C
′) ⊂ C ′
3. Γ(C ′) is closed in Γ(C ).
Then for any closed eigenbundle v ∈ H1(X,A ) with eigenvalue in K0 and at
least one divisor C′ ∈ Γ(C ′) the unique invariant divisor C ∈ Γ(C ) of v also
lies in Γ(C ′).
Proof. The proof is trivial since C = limk→∞(ΓfC )
k(C′)/λk where λ ∈ R>0 is
the eigenvalue of v.
4 Subsheaf Cohomology
In applications of Theorem 12 it is common that there is a well understood
exact sequence of sheaves
d0→ S1
d1→ S2
d2→ · · · (5)
and that B is a subsheaf of Sk for some k, A is the kernel of dk
: B →
Sk+1 and C is the image of B in Sk+1. Moreover, in these cases the self co-
homomorphism f on A → B → C is induced by an f self cohomomorphism
of the sequence (5). In order to apply Theorem 12 to these cases we need to
understand the R module H1(X,A ) and its induced self map.
There does not seem to be a computationally useful way to extract an
injective resolution of A using subsheaves of S0
d0→ S1
d1→ · · · even if this
last sequence is acyclic. Consider for example the case where for each n,
Sn is the sheaf of currents of degree n and B ⊂ Sk is a subsheaf of mildly
regular currents. It is not clear one could make the regularization method of
[dR84] work to compare H1(X,A ) to deRham cohomology groups because
his chain homotopy operator A does not restrict well to B since dA does not
preserve regularity. We use a standard sheaf cohomological trick, which we
include here as a proposition which we will need and which we expect to be
commonly used in conjuction with Theorem 12 because of the requirement
that Γ(B) be a Banach space.
Theorem 16 (Subsheaf Cohomology). Assume we are given an exact se-
quence of sheaves S0
d0→ S1
d1→ S2
d2→ · · · and that B is a subsheaf of Sk for
some k ≥ 1. Let A = ker dk
, and B′ be the preimage of B under dk−1.
Further assume that for each j ≥ 1 we have Hj(X,B′) = 0, Hj(X,B) = 0
and for any m satisfying 0 ≤ m ≤ k − 1 we have Hj(X,Sm) = 0 for j ≥ 1.
Then for each n ≥ 1 there is a canonical isomorphism
Hn(X,A ) ∼= Hn+k(X, ker d0).
Proof. While this result is essential for us, its proof is a standard cohomo-
logical trick. First one notes that ker dk−1
= ker dk−1 by the definition of
B′. One has the short exact sequences of sheaves:
ker dk−1 → B′ → (dk(B′) = A )
ker dj → Sj → ker dj+1, j = 0, . . . , k − 2.
Considering the long exact sequences for these shows that the induced maps
Hn(X,A )→ Hn+1(X, ker dk−1) andHn+j(X, ker dk−j)→ Hn+j−1(X, ker dk−j−1)
are isomorphisms for j = 1, . . . , k−1. Composing each of these canonical iso-
morphisms gives a canonical isomorphism fromHn(X,A )→ Hn+k(X, ker d0).
Remark. We take it as clear from the functorality of the δ map in the long
exact sequence that given an f -self cohomomorphism of S0
d0→ S1
d1→ S2
· · · which maps B to itself that the induced map of H1(X,A ) is identified
with the induced map of Hk+1(X, ker d0) via the above isomorphism.
We will need one more tool be able to make effective use of Theorem 16
for calculating sheaf cohomology of subsheaves of sheaves of currents.
Definition 17. By an interval flow h on a bounded open interval I ⊂ R we
will mean the flow obtained by integrating a vector field of the form σ(t) ∂
where σ is positive exactly on I and zero elsewhere. We use h(x, t) to denote
the location of x ∈ R after following the flow for time t.
Definition 18. By an n-box in Rn we will mean an open subset which is
a product of n bounded open intervals I1, . . . , In. By an n-box in an n
dimensional manifold we will mean an n-box which is compactly supported
in some coordinate patch. By an n-subbox of an n box U = I1 × · · · × In we
will mean an n box of the form I ′1 × · · · × I ′n where I ′k is a subinterval of Ik
for each k ∈ 1, . . . , n.
Definition 19. By an n-box flow we will mean the Rn action h on Rn
which is the product of n interval flows h1(t1), . . . , hn(tn) on Rn. That is
h(x, t) = (h1(x1, t1), . . . , hn(xn, tn)) where x = (x1, . . . , xn), t = (t1, . . . , tn)
and h1, . . . , hn are interval flows on I1, . . . , In respectively. We refer to the
n-box I1 × · · · × In as the open support of the n-box flow. We will often ht
to denote the diffeomorphism h(·, t) : Rn → Rn.
Definition 20. Let h be an n-box flow on an n-box B. Let ρ be a compactly
supported smooth volume form on Rn. With this data we define an operator
Sh,ρ on smooth k forms on any n box U containing B by
Sh,ρ(φ) =
h∗t (φ)ρ(t) (6)
We say Sh,ρ defines a box smear on U , or smears U . We will omit the
subscript from Sh,ρ when the meaning is clear from context. It is clear S(φ)
is compactly supported in U if φ is.
It is clear from the definition of S that if ψ is an n− k form on U then∫
SH,ρ(φ) ∧ ψ =
φ ∧ S−H,ρ(ψ)
where−H is the family Ht with the parameter negated. From this motivation
we define a smear of a current.
Definition 21. Given h, ρ defining a smear on an n box U we define the
smear Sh,ρ on currents on U via
< Sh,ρ(C), φ >≡< C,S−h,ρ(φ) > .
Lemma 22. Given h, ρ defining a smear S on an n box U then d
S(dC) for currents C on any open subset of U containing the open support
of the smear. Also, restricted to the open support of the smear, S(C) is a
smooth form on V .
Proof. We remark that it is clear that d
= S(dφ) for forms φ, and
consequently for currents φ via the definition.
Because on the open support of the smear, a smear is just convolution
with a smooth function, then we see that if V is an open subset of the open
support of smear S on U then for any current C on U , S(C)
is a smooth
form on V .
Proposition 23. Let B be a sheaf of degree k currents. Assume that B
contains the sheaf of smooth k forms on X, and that B(U) is closed under
smears on any n-box U ⊂ X. Let B′ be the preimage under d of B in the
sheaf of degree k − 1 currents. Then B′ is soft, and therefore, Γ-acyclic.
Proof. To show that B′ is soft it is sufficient to show that B′ is locally soft
([Bre97] page 69). Given an n-box U in X we therefore only need to show
that if K is a closed subset of X in U and if W is an open neighborhood
of K then given any member B′0 of B
′(W ) there is an open neighborhood
W0 ⊂ W of K and a member B′ ∈ B′(U) such that B′
= B′0
Choose any pair of open sets V1, V2 such that K b V1 b V2 b W . Then
V2 \ V1 is compact and can therefore be covered by finitely many (open) n-
subboxes Y1 . . . , YN of U . Moreover these subboxes can all be chosen to be
disjoint from K and to lie inside W . Letting S1, . . . ,Sn be smears on U with
open support Y1, . . . , YN respectively then let B = S1(S2(· · · (SN(B′0)) · · · )).
Then on each Yj, B is given by a smooth k form. Also, B
= B′0
. Finally,
we choose a smooth function ψ : U → [0, 1] which is one on a neighborhood
of V1 and zero on a neighborhood of U \ V2. Then the current B′ ≡ ψB
extends (by zero) to a current on all of U . Then for each Yj, B
smooth function times a smooth form. Thus d(B′
) is a smooth form and
Figure 1: A current comprised of parallel submanifolds smeared and cropped.
lies in B(Yj). The boxes Yj cover V2 \ V1. Outside V2, B′ is identically zero.
We know that dB ∈ B(W ) by Lemma 22. We also know that ψ ≡ 1 on
an open neighborhood W1 of V1. Thus d(B
) = d(B
) ∈ B(W1). We
thus conclude that B′ ∈ B′(U) since its restriction to each Yj, to W1 and to
U \ V2 is a section of B′. Letting W0 = V2 \ (Y1 ∪ Y2 ∪ · · · ∪ YN) then W0 is
an open neighborhood of K, then W0 ⊂ W1 so B′
= B′0 since
W0 is disjoint from the open support of each of the smears S1, . . . ,SN . This
completes the proof that B′ is soft.
The following gives a broad generalization of the equalivalence of the co-
homology of currents with the deRham cohomology groups. To the author’s
knowledge, this result is new.
Corollary 24. Let B be a sheaf of degree k currents. Assume that B con-
tains the sheaf of smooth k forms on X, and that B(U) is closed under
smears on any n-box U ⊂ X. Letting A be the subsheaf of d closed members
of B, then
Hm(X,A ) = Hm+k(X,K),
where K is R or C depending on whether or not we allow complex valued
currents and forms.
Proof. This is an immediate consequence of Proposition 23 and Theorem 16.
5 Invariant Currents
Notation 1. If G is some sheaf of functions on a smooth orientable manifold
X we will use F k(G ) to denote the sheaf of k forms on X with coefficients
in G . We will let F kc (G ) be the subsheaf of closed (in the sense of currents)
members of F k(G ).
It will be convenient to use either degree or dimension of a current de-
pending on the context (just as dimension and codimension are useful for
discussing manifolds), so we will not stick to just one of these terms. We will
let C k denote the sheaf of degree k currents with the index written above
as is typical for cohomology since d increases the degree. We will similarly
write Ck for the sheaf of dimension k currents with the index written below
since d decreases dimension as is common for homology. We use the following
convention to realize a form α as a current so that if α is C1 then dα is the
same whether computed as a current or a form.
Definition 25. Given an k form α with L1 coefficients on an n manifold X
we realize α as a degree k current via
β 7→ (−1)(
α ∧ β
Definition 26. Given a (possibly complex) nonzero deRham cohomology
class c ∈ HkdeRham(X) with f
∗(c) = α · c for some scalar α ∈ C we will refer
to a current C in the same cohomology class as α as an eigencurrent for f if
f ∗(C) = αC.
Currents naturally pushforward, rather than pullback. Because we are
considering maps which are not necessarily invertible we need to address
how this pullback is performed. If f has critical points it is impossible to
define a continuous pullback operation f ∗ on all currents in a way that agrees
with expected cases. For instance, consider f(x) = x2 and let Ca be the
dimension one current on R with Ca(h(x)dx) = h(a), i.e. Ca is a unit mass
vector. Then the pullback f ∗(Ca) should be the sum of weighted unit masses
at the two preimages of this vector (just like the pullback of a point mass
is a sum of point masses each weighted by multiplicity), that is, f ∗(Ca) =
C√a − C−√a
. However, these pullbacks do not converge to a current as
a → 0 so f ∗(C0) is not defined. Since we want f ∗ to be continuous, we are
forced to work with currents that have some extremely mild regularity. We
address this in the next section.
5.1 Nimble Forms and Lenient Currents
Finding a good set of currents to use to study smooth finite self maps (not
necessarily invertible) of compact manifolds turns out to be rather delicate.
Our solution is to first expand our class of forms to include pushforwards (in
the sense of currents) of forms through an appropriate class of smooth maps.
Then we restrict our attention to currents which act on this extended class
of forms.
This solution has the very nice property that it can potentially be adapted
directly to study the dynamics of other various other categories of smooth
maps (by simply changing which forms are considered nimble, according
to the class of maps used). It will convenient to first define the natural
pushforward operator on forms:
Definition 27. Given a compact orientable manifold X we let SX be the
category of smooth maps f : X → X of nonzero degree and having the
property that the critical set has measure zero. We use critical set here to
mean the points at which Df is not invertible.
It follows from our definition that the image of any set of positive measure
under some f ∈ SX has positive measure.
Definition 28. Given a compact orientable manifold X we define N k to be
those currents ϕ which are a finite sum of currents of the form p∗(σ) where
p : X → X is a map in SX and σ is a form of degree k. The pushforward
p∗(σ) is computed in the sense of currents.
We will later show that nimble forms are also, in fact, bona fide forms.
Definition 29. We topologize N k by saying ϕj → ϕ in N k if for sufficiently
large j there are maps f1, . . . , fk and k forms σ1j, . . . , σkj as well as forms
σ1, . . . , σk such that
i fi∗(σij) = ϕj and
i fi∗(σi) = ϕ (where pushfor-
wards are taken in the sense of currents) and for each i ∈ 1, . . . , k, the forms
σij converge to σi in the strong sense (i.e. all derivatives converge uniformly).
Lemma 30. Given a compact orientable manifold Y , N k(Y ) is a topological
vector space.
Proof. This follows easily from our definition of the topology.
We now define the corresponding space of currents.
Definition 31. We define the dimension k lenient currents Lk(Y ) to be
the topological dual of N k(Y ). Every member of Lk(Y ) is a dimension k
current, but with the added structure of its action on all nimble k forms. We
give Lk the weak topology, i.e. Ci → C in Lk iff < Ci, ϕ >→< C, ϕ > for
every ϕ ∈ N k. We write L k for the lenient currents of degree k.
We define operations of wedge products with smooth forms as is usual for
currents. It is clear that the lenient dimension k currents give a sheaf on X.
The following properties of nimble forms are also immediately clear.
Lemma 32. Let f : X → X be a member of SX . The pushforward (as
a current) of a nimble k form by f is again a nimble form. Moreover
f∗ : N
k(X)→ N k(X) is continuous (in the topology of nimble forms). Also
the exterior derivative of a nimble form (as a current) is a nimble form and
d : N k(X)→ N k+1(X) is continuous.
The basic necessary facts about pulling back lenient currents are then
immediate. We state them here:
Lemma 33. Given f : X → X a member of SX the induced map f ∗ on the
sheaf of lenient degree k currents is an f cohomomorphism of sheaves. Both
f ∗ : L k(X) → L k(X) and d : L k(X) → L k+1(X) are continuous. Lastly,
f ∗d = df ∗ : L k(Y )→ L k+1(X).
Proposition 34. Assume that f : X → X is a member of SX . Let R be the
regular set of f . By Sard’s theorem R has full measure. Since the critical set
is compact then R is an open subset of X. Since the preimage of a measure
zero set has measure zero for SX maps then f−1(R) is also a full measure
open set in X. There is a well defined operation f? which maps k forms on
f−1(R) to k forms on R. Given a k form β on X, f?(β) is defined on any
open subset V ⊂ R such that each component U1, . . . , Um of f−1(V ) maps
diffeomorphically onto V by the formula
f?(β)
deg f
(β) · σi (7)
where σi ∈ {±1} is the oriented degree of f
: Ui → V . The pushforward
f? satisfies:
• f?d = df? (keeping in mind that f? returns a current on R)
• f?(1) = 1
• f?(f ∗(β) ∧ α) = β ∧ f?(α)
• (f?)n = (fn)?
• The formula ∫
f ∗(β) ∧ α =
β ∧ f?(α) (8)
holds for any k form β with L∞loc coefficients on Y and any smooth n−k
form α on X. This justifies using f? to pullback currents. (Part of the
conclusion is that both sides are integrable.)
Proof. Each statement is a consequence of formula (7) except the integrabil-
ity conclusion for equation (8). Local charts can be given which are bounded
subsets of Rn and for which Df remains uniformly bounded (over each of the
charts) and thus f ∗(β) will be a form with L∞loc coefficients in these charts.
Thus the left hand side of (8) is the integral of a bounded function over a
finite union of bounded charts and is therefore absolutely integrable. Since
∗(β) ∧ α) = β ∧ f?(α) it is sufficient to show that if γ is an n form with
L∞loc coefficients then ∫
f−1(R)
f?(γ). (9)
Typicaly f?(γ) is unbounded so we need to show that the right hand side of
(9) is integrable. About any point x ∈ R we can find an open V such that
each of the preimages U1, . . . , Uk of V is mapped diffeomorphically onto V .
Since X is orientable and n dimensional there is a well defined notion of the
absolute value of an n form. Then∫
|f?(γ)| ≤
deg f
((f
 = ∑
|γ| =
f−1(V )
NowR is covered by countably many such sets V and listing them as V0, V1, V2, . . . ,
we can let V ′0 = V0, V
1 = V1 \V0, V ′2 = V2 \ (V0∪V1), . . . . Then R is the union
of the countable collection of disjoint measurable sets V ′j and∫
|f?(γ)| =
|f?(γ)| ≤
f−1(Vj)
|γ| =
f−1(R)
Since
f−1(R)
|γ| is finite then f?(γ) is an L1 form. Using precisely the same
argument but with the absolute values removed and the inequalities replaced
with equalities then shows
f?(γ) =
f−1(R)
Since R and f−1(R) are open and full measure then f? is an operator
which takes in forms on X and returns forms defined almost everywhere on
We now show that nimble forms are bona fide forms.
Lemma 35. If g : X → X is a map in SX and σ is a smooth k form on X
then the current g∗(σ) is the current of integration against the form g?(σ).
Proof. If ϕ is a smooth n − k form then by definition < g∗(σ), ϕ >=<
σ, g∗(ϕ) >= (−1)(
σ ∧ g∗(ϕ) = (−1)(
g?(σ) ∧ ϕ =< g?(σ), ϕ >
by formula (8) of Proposition 34
As described in [Fed69], an inner product on a vector space V can be
viewed as an isomorphism ` : V → V ∗ satisfying certain properties. The
inverse of ` gives the induced inner product on V ∗. The fact that < v,w >≤
‖v‖ · ‖w‖ with equality iff v and w are scalar multiples implies that the inner
product norm on V ∗ is the same as the operator norm of V ∗ acting on V .
The induced map
V ∗ gives an inner product on
We call this the canonical inner product on
V induced by the inner prod-
uct on V . Hence, given a Riemannian metric on X, there are canonical
smoothly varying inner products on
TxX and
T ∗xX for each x ∈ X.
At any point x ∈ X we define ‖
Dxf‖ to be the operator norm of the
linear function
Dxf :
TxX →
Tf(x)X. We define ‖
Df‖ to be
the L∞loc norm of the map x 7→ ‖
Dxf‖. Also, given a k form ϕ we define
the comass ‖ϕ‖L∞loc of ϕ to be the L
loc norm of the function x 7→ ‖
ϕx‖. It
is clear that the k forms with the comass norm is a Banach space. We now
show that the k forms with L∞loc coefficients are naturally lenient currents.
We start by defining the action on nimble forms.
Definition 36. Given an n− k form C with L∞loc coefficients we define
< C, p∗(σ) >= (−1)(
n−k+1
C ∧ p?(σ)
Lemma 37. The space F n−k(L∞loc) of n−k forms with L
loc coefficients under
the comass norm includes continuously into Lk(X) where the action of C ∈
F n−k(L∞loc) on some ϕ =
i fi∗(σi) ∈ N
k(X), with each fi ∈ SX and each
σi ∈ F k(C∞) is given by
< C, ϕ >≡
f ∗i (C) ∧ σi.
Proof. The assumption that X is compact means that any two Rieman-
nian metrics on X are comparable. Choose one so the notion of the comass
norm makes sense. The result is then a straightforward consequence of equa-
tion (8), Lemma 35, and our definitions.
Remark. It follows that a current with local F k(L∞loc) potentials is also a
lenient current.
Remark. Given a member C of F k(L∞loc) then f
∗(C) is the same whether
done as a lenient current or as a form. This, along with the fact that df ∗ =
f ∗d justifies the ad hoc pullback of closed positive (1, 1) currents used so
successfully in holomorphic dynamics. Similarly dC gives the same result
whether calculated as a lenient current or a form if C ∈ F k(C1).
5.2 Hölder Lemmas
We will want to apply Corollary 15 to show that each eigencurrent we con-
struct has local d potentials (or ddc potentials in the holomorphic case) which
are forms with Hölder continuous coefficients. In order to do this we will need
a few facts which we include here in order to avoid having to include regu-
larization results as afterthoughts to our main theorems.
Observation. Let Hα be the functions with coefficients that are Hölder of
exponent at least equal to some fixed α > 0. Since diffeomorphisms preserve
Hölder exponents and averages of Hölder functions are Hölder then we take
it as clear that Corollary 24 applies to show that H1(X,A ′) = H1(X,A )
where A ′ is the closed members of F k(Hα)) and A is the closed degree k
currents.
Lemma 38. Let X be a compact manifold (real or complex) with a Rieman-
nian metric and of real dimension n. Let f : X → X be a smooth map. Then
local coordinate charts Ui can be chosen on X (each representing a convex
open subset of Rn) so that there is a positive constant 1 < M so that for any
k form ϕ, there exist constants c, C > 0 such that writing each fk∗(ϕ) in any
of the charts Ui as
fk∗(ϕ) =
akidx
then each function aki satisfies
|aki| ≤ c · ‖fk∗(ϕ)‖comass (10)
and for each j ∈ 1, . . . , n,
∂aki
 ≤ C ·Mk.
Proof. Equation (10) is a basic fact.
The rest is a straightforward consequence of realizing a self map of a
manifold as being made up of a bunch of maps between different coordinate
patches in Rn. That is, one chooses an open cover of patches Ui of X. Each
patch is realized in Rn as a round ball. Thinking of each patch as lying in Rn
then we can find explicit maps from between open subsets of Rn of the form
pij : Ui ∩ f−1(Uj)→ Uj. By shrinking each open ball Ui a small amount the
resulting patches still cover X but the derivatives of the maps pij are all now
bounded (since we are working on relatively compact subsets of the previous
maps pij).
Then given any x we can keep track of which patch fk(x) is in at each time
and can then realize the map fk(x) as a composition pi1i2 ◦ pi2i3 ◦ · · · ◦ pik−1ik .
Since each partial derivative of each pij is uniformly bounded then any partial
derivative of the composition grows at most exponentially with k and we are
done.
The following observation will also be useful:
Lemma 39. If there are positive constants c, C,m,M with m < 1 < M
such that a sequence of smooth functions hk on an open convex set U ⊂ Rn
satisfies
‖hk‖sup < c ·mk
and www∂hk
< C ·Mk
for all k ∈ 0, 1, 2, . . . then h1+h2+h3+. . . converges to a bounded continuous
function which is Hölder of any exponent α <
log(m)
log(m/M)
Proof. The proof is elementary.
5.3 Eigencurrents for Cohomologically Expanding Smooth
We will call a section V of
TX a k-vector field. We define ‖V ‖L∞loc to be
the L∞loc norm of the function x 7→ ‖Vx‖. Whether Theorem 12 applies to a
map will depend the size of ‖
Df‖. Replacing f with an iterate does not
affect the needed estimate so we make the following definition.
Definition 40. We define Υk to be the limit supremum as j → ∞ of
D(f j)‖
j . It follows that Υ1 ≥ eλ for any Lyapunov exponent λ and
that Υk ≤ Υk1 ([Fed69] page 33).
We let B be the sheaf F k−1(L∞loc). The norm ‖ · ‖∞ clearly makes Γ(B)
into a Banach space. Given a member B ∈ Γ(B), since the operator norm on
TxX is equal to the norm already defined on
T ∗xX for each x ∈ X
then ‖B‖∞ is equal to supremum of the L∞loc norm of the function x 7→ B(Vx)
as V varies over all L∞loc k-vector fields of norm no more than one.
Theorem 41. Given f : X → X an a map in SX for the compact orientable
manifold X, assume that c ∈ HkdeRham(X) is a cohomology class (using ei-
ther real or complex deRham cohomology) which is an eigenvector for f ∗ with
eigenvalue β. Assume also that |β| > Υk−1. Then there exists a unique eigen-
current C with local F k−1(L∞loc) potentials representing the class c. Moreover
C has local F k−1(H) potentials.
Also, given any neighborhood U ⊂ X of any point in the support of C,
then for every lenient current C′ with local F k−1(L∞loc) potentials and which
represents the cohomology class c then fk(U) ∩ Supp C′ 6= ∅ for all large k.
Assume that the linear map f ∗ : HkdeRham(X)→ H
deRham(X) is dominated
by a single simple real eigenvalue r. Given C′ any current which has local
F k−1(L∞loc) potentials and which represents a cohomology class in the Υ
chronically expanding subspace of HkdeRham(X), then the successive rescaled
pullbacks fk∗(C′)/rk of C′ converge to a multiple of C in the sense of lenient
currents (and thus also in the sense of currents).
Proof. We let B = F k−1(L∞loc), A and C be the kernel and image respec-
tively of B
d→ L k. By Theorem 24, H1(X,A ) can be canonically identified
with Hk(X,K). Since B is Γ-acyclic then every member of H1(X,A ) is a
closed bundle with respect to the short exact sequence A → B → C .
From Lemma 33 there is an induced f cohomorphism of the short ex-
act sequence A
ι→ B d→ C . Also Γ(C ) is a space of lenient currents by
Lemma 33 and thus has a natural structure as a topological vector space. If
a sequence Bi ∈ Γ(B) converges to B ∈ Γ(B) then < dBi, ϕ >=
Bi∧dϕ =∫
B ∧ dϕ =< dB, ϕ > so the map d : Γ(B)→ Γ(C ) is continuous.
The cohomomorphism ΓfB is pullback f
∗ of differential forms. Fixing any
real α satisfying Υk−1 < α < |β| it is clear from the definition of Υk−1 that one
can choose a real d > 0 such that ‖
D(f `)‖ ≤ d ·α` for all ` ∈ N. The `th
pullback f `∗(B) of B ∈ Γ(B) satisfies ‖f `∗(B)‖∞ = supV ‖B(
D(f `)(V ))‖∞
where the supremum is taken over all k-covector fields V with ‖V ‖∞ ≤ 1.
However
D(f `)(V ) is a k-covector field of norm no more than ‖
D(f `)‖,
so ‖f `∗(B)‖∞ ≤ ‖B‖∞ ·
D(f `)‖∞ ≤ d · α`‖B‖∞.
Given any W in the Υk−1 chronically expanding subspace of H
k(X,K),
we can alter our choice of α > Υk−1 so that W also lies in the α chronically
expanding subspace of Hk(X,K).
We can therefore apply Theorem 12 to conclude that there is a (unique)
map κ : W → Γ(C ) such that f ∗κ = κf ∗, where the first f ∗ is pullback of
currents and the second is pullback on Hk(X,K).
In fact κ(W ) lies in the space of currents with locally Hölder potentials
(meaning F k−1(H) potentials) by applying Corollary 15 in conjunction with
Observation 5.2, Lemma 38 and Lemma 39. The second half of the Theorem
is a consequence of equation (3).
Remark. Theorem 41 gives regular degree one eigencurrents for every eigen-
value of f ∗ : H1(X,K) → H1(X,K) of norm greater one without requiring
any constraints on the local behavior of f . The degree one eigencurrents
seem to be, in some sense, more robust than currents of lower dimension,
including invariant measures. Moreover since codimension one closed sub-
manifolds are closed currents with local F 0(L∞loc) potentials then successive
rescaled preimages of such manifolds in the right cohomological class will
converge to the eigencurrent.
Remark. The fact that eigencurrents constructed via Theorem 41 have local
potentials which are forms does not imply their support has positive Lebesgue
measure as the classical example of a monotonic nonconstant function which
is constant on a set of full measure shows.
Remark. The assumption that f ∗ : H1deRham(X)→ H
deRham(X) is dominated
by a single simple real eigenvalue r is not essential, but just meant to handle
the simplest case. In fact the proof actually shows that if W lies in the
Υk−1 chronically expanding subspace of H
k(X,K) then every current in the
invariant plane κ(W ) ⊂ Γ(C ) of currents has local F k−1(H) potentials and
any current with cohomological class in W with local F k−1(L∞loc) potentials
is attracted to κ(W ) under successive rescaled pullback.
Since measures are of particular interest in dynamics, we note thatH1(X,F n−1(L∞loc)) =
Hn(X,K) = K by Corollary 24 so there is a unique f ∗ eigenvalue and it is
precisely the topological degree of f . We thus obtain:
Corollary 42. Given that Υn−1 < deg f then there is a unique dimension
zero eigencurrent C with F k−1(L∞loc) potentials (and in fact it has F
k−1(H)
potentials) and the successive rescaled preimages of any C′ with F k−1(L∞loc)
potentials converge to C. If additionally there is no point x ∈ X about which
f is locally an orientation reversing diffeomorphism then C (and every other
member of κ(W )) is a positive distribution and is therefore a Radon measure.
Proof. Since f ∗ pulls back dimension zero currents (i.e. distributions) which
are positive to distributions which are positive then by Corollary 3.1 the
distribution C is positive. It is therefore a Radon measure (see e.g. [HL99]
page 270).
Remark. In the case where f is orientation reversing on some parts of X (but
not on all of X) some special remarks apply. If it happens that successive
rescaled images of some point converge to a dimension zero eigencurrent then
since preimages of points are counted with multiplicity then when pulled back
through a portion of X on which f reverses orientation the sign of a point
is flipped. Thus in this case the eigencurrent may not describe so much
the distribution of preimages as the relative density of preimages counted
negatively as compared to those counted positively. The number of actual
preimages of a point may grow exponentially faster than the degree of the
map in such cases so that dividing by the degree does not yield a measure in
the limit unless some such “cancellation” takes place in the limit. One would
expect that the corresponding eigencurrents have local potentials which are
not of bounded variation in such a case.
5.4 Eigencurrents for Smooth Covering Maps
We will call a covering map which is locally a diffeomorphism a smooth
covering map. We now consider the special case of smooth self covering
maps f : X → X of a compact smooth orientable manifold X. We show
that in this case we have a substantially broader collection of currents whose
successive pullbacks converge to an eigencurrent, albeit we need different
estimates for Theorem 12 to apply. We will pull back currents by pushing
forward forms with f?. Since the regular set of f is all of X then f? is a well
defined operator from smooth forms to smooth forms.
Definition 43. For a map satisfying the hypothesis of Proposition 34 we
define the operation f ∗ from currents on X to currents on Y by
< f ∗(C), α >≡< C, f?(α) > .
Clearly f ∗ preserves the dimension of a current.
Let Mk−1 be the sheaf for which Mk−1(U) is the Banach space of bounded
linear operations on the topological vector space comprised of F k−1(C∞)(U)
with the ‖ · ‖∞ norm. Equivalently, Mk−1 is the sheaf of dimension k − 1
currents of finite mass.
Choose a Riemannian metric on X. If f : X → X is a smooth cover
then for each x ∈ X and each ` ∈ N, Dx(f `) : TxX → Tf`(x)X is invertible.
We let νk(x, `) be the operator norm of the inverse of
TxX →∧k
Tf`(x)X. We define νk(`) = supx∈X νk(x, `)
1/`. We define νk = lim sup`→∞ νk(`).
The iterated pushforward operation f `? : F
k−1(C∞)(X) → F k−1(C∞)(X)
satisfies ‖f `?(ϕ)‖∞ ≤ νk(`) · ‖ϕ‖∞ as is straightforward to verify. If f is in-
vertible then νk is a bound on the growth of the k
th wedge product of the
derivative under f−1. For non-invertible f , νk represents a bound on the
growth of the kth wedge product of the derivative under any sequence of
successive branches of f−1.
Theorem 44. Given f : X → X a smooth self covering map and that
c ∈ HkdeRham(X) is a cohomology class (using either real or complex deR-
ham cohomology) which is an eigenvector for f ∗ with eigenvalue β. Assume
also that |β| > νk−1. Then there exists a unique eigencurrent C with local
Mk−1 potentials representing the class c. Moreover C has local F
k−1(C0)
potentials. Consequently C is a current of order one.
Also, given any neighborhood U ⊂ X of any point in the support of C, then
for every lenient current C′ with local Mk−1 potentials and which represents
the cohomology class c then fk(U) ∩ Supp C′ 6= ∅ for all large k.
Assume that the linear map f ∗ : HkdeRham(X)→ H
deRham(X) is dominated
by a single simple real eigenvalue r. Given C′ any current which has local
Mk−1 potentials and which represents a cohomology class in the ν
k−1 chroni-
cally expanding subspace of HkdeRham(X), then the successive rescaled pullbacks
fk∗(C′)/rk of C′ converge a multiple of C.
Proof. We let A and C be the kernel and image respectively of d : Mk−1 →
C k. Since df? = f?d then pullback of currents gives an f cohomomorphism
of the short exact sequence of sheaves A →Mk−1 → C .
Since ΓMk−1 is the continuous linear operators on a normed vector space
then it is a Banach space. From the observations previous to the statement
of Theorem 44 one concludes that for any α > νk−1 there is a constant d > 0
such that ‖f `∗(B)‖ ≤ d · αk‖B‖ for all ` ∈ N.
Since Γ(C ) is a space of currents it is naturally a topological vector space
over K.
The map f ∗ : Γ(C ) → Γ(C ) is continuous since if Ci → C in Γ(C ) then
< f ∗(Ci), ϕ >=< Ci, f?(ϕ) >→< C, f?(ϕ) >=< f ∗(C), ϕ >.
If Pi → P in ΓMk−1 (using the Banach space structure) then ‖Pi−P‖ →
0 by assumption then ‖P(dϕ) − Pi(dϕ)‖ ≤ ‖P − Pi‖ · ‖dϕ‖ → 0. Hence
< dPi, ϕ >= Pi(dϕ)→ P(dϕ) =< dP, ϕ > and so we conclude that the map
d : ΓMk−1 → Γ(C ) is continuous.
Given any W in the νk−1 chronically expanding subspace of H
k(X,K),
we can alter our choice of α > νk−1 so that W also lies in the α chronically
expanding subspace of Hk(X,K).
We can therefore apply Theorem 12 to conclude that there is a (unique)
map κ : W → Γ(C ) such that f ∗κ = κf ∗, where the first f ∗ is pullback of
currents and the second is pullback on Hk(X,K).
In fact κ(W ) in the currents with locally continuous potentials by apply-
ing applying Corollary 15 in conjunction with Observation 5.2, Lemma 38
and Lemma 39. The second half of the Theorem is a consequence of equa-
tion (3).
Proposition 45. Let Y be an oriented codimension k submanifold of X. If
the cohomological class of Y (as a current) lies in the νk−1 chronically expand-
ing subspace of Hk(X,K) then the successive rescaled preimages of Y con-
verge to the invariant plane of currents κ(W ). If f ∗ : Hk(X,K)→ Hk(X,K)
is dominated by a single real eigenvalue r > νk−1 then the successive rescaled
preimages of Y converge to a multiple (possibly zero) of the r eigencurrent.
In particular, if νn−1 < deg f then the successive rescaled preimages of any
point converge to the unique invariant measure with Mn−1 potentials.
Proof. This follows immediately from Theorem 44 if we show that Y has
local potentials in Mk−1. This is equivalent to showing that locally Y = dP
where < P,ϕ >≤ a · ‖ϕ‖∞ for some a > 0. Let B be a ball in Rn and Y0
a k-plane in Rn. Then there is a k + 1 half plane P such that, as currents
in U , ∂P = Y0. Moreover it is clear that < P,ϕ >≤ a‖ϕ‖∞ for some real
a > 0. (There are also local potentials for Y which are given by forms with
L1loc coefficients. These can be constructed by choosing a projection π from
U \ Y0 to a codimension one cylinder C with axis Y0, and choosing a volume
form σ on C. The local potential is the pullback π∗(σ).)
Remark. As with Theorem 41, Theorem 44 gives regular degree one eigencur-
rents for every eigenvalue of f ∗ : H1(X,K)→ H1(X,K) of norm greater one
without requiring any constraints on the local behavior of f . In holomorphic
dynamics much progress has been made in constructing degree one eigencur-
rents and then constructing dynamically important invariant measures via
a generalized wedge product (see the references cited at the beginning of
Section 6).
Remark. The proof of Proposition 45 could clearly be modified to apply to
many singular manifolds as well.
6 Holomorphic Endomorphisms
We now restrict our interest to holomorphic dynamics. Thus all manifolds
are assumed to be complex manifolds and all maps are assumed to be holo-
morphic unless stated otherwise.
Holomorphic endomorphisms of the Riemann sphere have been studied
in great detail. For endomorphisms much of the theory is still in its be-
ginnings. Much attention has been paid to holomorphic automorphisms of
C2 [FM89], [FS92], [HOV94], [HOV95], [BS91a], [BS91b], [BS92], [BLS93],
[BS98a], [BS98b], [BS99] or K3 surfaces [Can01], [McM02], the major de-
velopments for endomorphisms have been on Pn, [FS94a], [FS94b], [FS95b],
[FS01], [FS95a], [JW00], [FJ03], [Ued94], [Ued98], [Ued97]. Recent signifi-
cant developments have been made for endomorphisms of Kahler manifolds
in [DS05]. The paper [DS05] shows existence of eigencurrents (or Green’s
currents) for endomorphisms of Kahler manifolds under a simple condition
on the comparative rates of growth of volume in two different dimensions.
They also show that a specific weighted sum of an arbitrary closed positive
smooth current will converge to the Green’s current, and that the Green’s
current has a Hölder continuous potential. In this setting our theorem shows
that arbitrary (rescaled) preimages of a broader class of currents will con-
verge to the Green’s current. A wide variety of results have been proven
in these various circumstances either showing the existence of invariant cur-
rents, showing convergence of currents to invariant currents, or studying the
properties of these invariant currents. We include here results that follow
from the method of this paper, which we are sure substantially overlap with
existing results. Presumably our cohomologicaly lifting theorem could be
used in conjuction with Theorem 12 to show existence of higher degree (k, k)
currents given certain bounds on local growth rates.
6.1 ddc Cohomology
Let Z be a complex manifold and let f : Z → Z be a holomorphic self map
of Z. Let H be the sheaf of pluriharmonic functions, let L∞loc be the sheaf
of locally bounded functions, and let C be the sheaf of currents with local
potentials in L∞loc, i.e. currents locally of the form dd
cb, for b a locally bounded
function. The members of C are closed (1, 1) currents on Z.
Using the usual pullback on functions, and the induced pullback on cur-
rents with function potentials (i.e. pullback the current by pulling back its
local potentials), then we get a self cohomomorphism of the exact sequence
of sheaves
H → L∞loc
ddc→ C . (11)
We note that H1(Z,H ) is a finite dimensional R vector space as can be
seen from the long exact sequence for the short exact sequence R→ O →H
where the first map is inclusion and the second takes the imaginary part. The
terms H1(Z,O) → H1(Z,H ) → H2(Z,R) give the finite dimensionality
since O is a coherent analytic sheaf (see e.g. [Tay02] page 302).
Then from Theorem 12 we obtain:
Corollary 46. Given v any closed eigenbundle of H1(Z,H ) for f ∗ with
eigenvalue r > 1, there is a unique closed (1, 1) current C such that limk→∞ f
k∗(C′)/rk
converges to C for any divisor C′ of v.
Remark. We note that the terms “closed eigenbundle” and “divisor” in Corol-
lary 46 are understood using the long exact sequence for (11).
We can apply Corollary 15 to show that
Corollary 47. Any such invariant current C so obtained has Hölder contin-
uous local potentials.
Proof. The result follows from Lemma 5.2, Lemma 38, the fact that the
ddc closed Hölder continuous functions are the same as the ddc closed L∞loc
functions and from Corollary 15.
Also from Observation 3.1,
Corollary 48. If v has a plurisubharmonic section the current C is positive.
7 Result via Invariant Sections
We stated early on that our construction of invariant members of H0(C ) for
a self cohomomorphism of a short exact sequence A → B → C of sheaves
could be done in terms of finding invariant sections of bundles. We illustrate
this here in a specific case where we can take advantage of geometry to make
further conclusions. Finding an invariant section of a bundle is equivalent to
finding an invariant trivialization of the bundle, and we will make our initial
statement in terms of a trivialization.
Let Z be a compact complex manifold. Let f : Z → Z be a holomorphic
endomorphism. Let p ∈ H1(Z,H ) be an eigenvector for f ∗ with real eigen-
value λ of norm greater than one. If f ∗ were to have complex eigenvalues of
interest, an analogous construction can be made to the one that follows.
We note that there is a canonical bundle map f̃ : f ∗(p) → p which gives
the map f on the base space. It is easy to show that there is a map σ : p→ λp
which is the identity on the base space and takes the form r 7→ λr+ b on the
fibers, where b is a constant. What is more, the map τλ is easily seen to be
unique up to the addition of a constant. Then define the map f̌ : p → p to
be the composition of
τλ→ λp = f ∗(p) f̃→ p.
Then f̌ is the map f on the base space and takes the form r 7→ λr + b on
the fibers.
Since every pluriharmonic bundle is trivial as a smooth bundle, then we
can choose a smooth trivialization t : p→ R, i.e. t(a+ r) = σ(a) + r for any
a ∈ p, r ∈ R, where a+ r is computed in the fiber containing a.
Theorem 49. There is a unique continuous trivialization g : p → R such
that:
g(a+ r) = g(a) + r for a ∈ p and r ∈ R,
g(f̌(a)) = λ · g(a) for a ∈ p,
moreover
g = lim
λ−k ◦ t ◦ f̌ ◦k
and the limit converges uniformly. Finally, the zero set of g is the image of a
section g : Z → p and is exactly the set of points whose forward image under
f̌ remains bounded.
Proof. Define a function T : p→ R by
T (a) ≡ t
f̌(a)
− λ · t(a).
Note that T descends to a well defined continuous function T : Z → R since
for an arbitrary r ∈ R one has T (a + r) = t
f̌(a + r)
− λ · t(a + r) =
t(f̌(a) + λr)− λ · (t(a) + r) = T (a).
One notes that since the function T is necessarily bounded if Z is compact
then defining
g(a) ≡ t(a) + λ−1 · T (a) + λ−2T (f̌(a)) + λ−3T (f̌ ◦2(a)) + · · ·
gives a continuous function g : p→ R satisfying the above two properties.
Assume g1 and g2 are two such functions. Then ∆ ≡ g1− g2 : p→ R is a
function satisfying ∆(a+ r) = ∆(a) for a ∈ p and r ∈ R so ∆ descends to a
continuous function ∆: p→ R satisfying ∆(f̌(a)) = λ ·∆(a). However since
λ > 1 one concludes that this is only possible if ∆ ≡ 0 since M is compact
so ∆(M) has compact image in R.
It is easy to check using the definition of T that λ−k ◦ t ◦ f̌ ◦k(a) is exactly
a partial sum of the first k terms of the above series and this gives the
convergence result. The conclusion about the section g is trivial.
The above construction can be carried through almost without modifi-
cation for any subspace of H1(Z,H ) on which f ∗ is expanding. This gives
an alternate way of understanding the convergence of preimages of sections.
The point is that if s is any section of p, i.e. the potential of a current C,
then 1
f ∗(C) is a current with potential which is the setwise preimage of s
under f̌ (this is easy to confirm from the construction of f). The Green’s
trivialization g shows that f̌ is uniformly repelling away from the image of
the invariant section g. Thus as long as s is bounded in p, (not even neces-
sarily continuous), then the successive preimages of s will converge uniformly
to the section g. Since uniform convergence of potentials implies convergence
of currents then the rescaled pullbacks of a current C converge to the cur-
rent with potential g. We already have this as a theorem, so we have not
restated it as such here. This is just an alternative approach. Note that in
the case where Z = P2 [FJ03] has given far more precise control of when the
successive rescaled preimages of a current will converge to the eigencurrent.
7.1 Sections version with an Invariant Ample Bundle
It is also interesting to consider the special case where there is an invariant
ample bundle with eigenvalue λ ≥ 2 an integer. Without loss of generality
we assume ` is very ample. The morphism of sheaves log | · | : O∗ → H
induces a map from holomorphic line bundles to pluriharmonic bundles. We
let p = log |`| be the corresponding pluriharmonic bundle.
It is easy to see that there is a holomorphic map ` → `λ which is of the
form σλ : z 7→ azλ, a ∈ C∗ on each fiber and is the identity on the base
space. There is also a canonical holomorphic map f̃ : f ∗(`) → ` which is a
line bundle map and is f on the base space.
One then defines the holomorphic map f̆ : `→ ` which is the composition
σk→ `k = f ∗(`) f̃→ `.
This map is of the form z 7→ azk on each fiber and is equal to the map
f : Z → Z on the base space. Let `∗ denote ` with its zero section removed,
so that log | · | : ` → p is a well defined continuous map. Since the preimage
of the zero section of ` under f̆ is the zero section then f̆ is a holomorphic
self map of `∗. It is easy to confirm that f̆ : ` → ` can be rescaled so that
the diagram
log |·|
log |·|
commutes.
Our Greens trivialization g : p→ R can be pulled back to give a Green’s
function G : `∗ → R on the punctured bundle `∗. It satisfies G(f̃(w)) =
λ · G(w) and G(βw) = G(w) + log |β| for w ∈ ` and β ∈ C∗. Since g is
a trivialization of an R bundle over a compact space, g is proper. Since
log | · | : `∗ → p is proper then G is proper. Thus, in this setting one can
construct a Greens function that is exactly analogous to the Green’s function
constructed on Cn+1 for a holomorphic endomorphism of Pn. Potentially one
could take advantage of the special geometry of very ample bundles to get
information about the dynamics in this situation.
8 Bibliography
References
[BLS93] Eric Bedford, Mikhail Lyubich, and John Smillie. Polynomial dif-
feomorphisms of C2. IV. The measure of maximal entropy and lam-
inar currents. Invent. Math., 112(1):77–125, 1993.
[Bre97] Glen E. Bredon. Sheaf Theory. Springer-Verlag, 1997.
[BS91a] Eric Bedford and John Smillie. Polynomial diffeomorphisms of C2:
currents, equilibrium measure and hyperbolicity. Invent. Math.,
103(1):69–99, 1991.
[BS91b] Eric Bedford and John Smillie. Polynomial diffeomorphisms of C2.
II. Stable manifolds and recurrence. J. Amer. Math. Soc., 4(4):657–
679, 1991.
[BS92] Eric Bedford and John Smillie. Polynomial diffeomorphisms of C2.
III. Ergodicity, exponents and entropy of the equilibrium measure.
Math. Ann., 294(3):395–420, 1992.
[BS98a] Eric Bedford and John Smillie. Polynomial diffeomorphisms of
C2. V. Critical points and Lyapunov exponents. J. Geom. Anal.,
8(3):349–383, 1998.
[BS98b] Eric Bedford and John Smillie. Polynomial diffeomorphisms of C2.
VI. Connectivity of J . Ann. of Math. (2), 148(2):695–735, 1998.
[BS99] Eric Bedford and John Smillie. Polynomial diffeomorphisms of C2.
VII. Hyperbolicity and external rays. Ann. Sci. École Norm. Sup.
(4), 32(4):455–497, 1999.
[Can01] Serge Cantat. Dynamique des automorphismes des surfaces K3.
Acta Math., 187(1):1–57, 2001.
[dR84] Georges de Rham. Differentiable Manifolds. Springer-Verlag, 1984.
[DS05] Tien-Cuong Dinh and Nessim Sibony. Green currents for holomor-
phic automorphisms of compact kähler manifolds. J. Amer. Math.
Soc., 18(2):291–312, 2005.
[Fed69] Herbert Federer. Geometric Measure Theory. Springer-Verlag,
1969.
[FJ03] Charles Favre and Mattias Jonsson. Brolin’s Theorem for Curves
in Two Complex Dimensions. Ann. Inst. Fourier, 53:1461–1501,
2003.
[FM89] Shmuel Friedland and John Milnor. Dynamical properties of
plane polynomial automorphisms. Ergodic Theory Dynam. Sys-
tems, 9(1):67–99, 1989.
[FS92] John Erik Fornaess and Nessim Sibony. Complex Hénon mappings
in C2 and Fatou-Bieberbach domains. Duke Mathematical Journal,
65(2):345–380, 1992.
[FS94a] John Erik Fornaess and Nessim Sibony. Complex dynamics in
higher dimension. In Complex Potential Theory, pages 131–186,
1994.
[FS94b] John Erik Fornaess and Nessim Sibony. Complex dynamics in
higher dimension. I. Astérisque, 222:201–231, 1994.
[FS95a] John Erik Fornaess and Nessim Sibony. Classification of recurrent
domains for some holomorphic maps. Math. Ann., 301(4):813–820,
1995.
[FS95b] John Erik Fornaess and Nessim Sibony. Complex dynamics in
higher dimension. II. In Modern methods in complex analysis
(Princeton, NJ, 1992), pages 135–182. Princeton Univ. Press,
Princeton, NJ, 1995.
[FS01] John Erik Fornæss and Nessim Sibony. Dynamics of p2 (examples).
In Laminations and foliations in dynamics, geometry and topology
(Stony Brook, NY, 1998), pages 47–85. Amer. Math. Soc., Provi-
dence, RI, 2001.
[GR84] H. Grauert and R. Remmert. Coherent Analytic Sheaves. Springer-
Verlag, 1984.
[Har77] Robin Hartshorne. Algebraic Geometry. Springer-Verlag, 1977.
[HL99] Francis Hirsche and Gilles Lacombe. Elements of Functional Analy-
sis, volume 192 of Graduate Texts in Mathematics. Springer-Verlag,
1999. Translated from the 1997 French original by Silvio Levy.
[HOV94] John H. Hubbard and Ralph W. Oberste-Vorth. Hénon mappings
in the complex domain. I. The global topology of dynamical space.
Inst. Hautes Études Sci. Publ. Math., (79):5–46, 1994.
[HOV95] John H. Hubbard and Ralph W. Oberste-Vorth. Hénon mappings
in the complex domain. II. Projective and inductive limits of poly-
nomials. In Real and complex dynamical systems (Hillerød, 1993),
volume 464 of NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., pages
89–132. Kluwer Acad. Publ., Dordrecht, 1995.
[JW00] Mattias Jonsson and Brendan Weickert. A nonalgebraic attractor
in P2. Proc. Amer. Math. Soc., 128(10):2999–3002, 2000.
[Kat03] Anatole Katok. Combinatorial constructions in ergodic theory and
dynamics, volume 30 of Unversity Lecture Series. American Math-
ematical Society, Providence, RI, 2003.
[McM02] Curtis T. McMullen. Dynamics on K3 surfaces: Salem numbers
and Siegel disks. J. Reine Angew. Math., 545:201–233, 2002.
[Tay02] Joseph L Taylor. Several Ccomplex Variables with Connections
to Algebraic Geometry and Lie Groups. American Mathematical
Society, 2002.
[Ued94] Tetsuo Ueda. Fatou sets in complex dynamics in projective spaces.
J. Math. Soc. Japan, 46:545–555, 1994.
[Ued97] Tetsuo Ueda. Complex dynamics on Pn and kobayashi metric.
In Complex dynamical systems and related areas, pages 188–191.
Surikaisekikenkyusho Kokyuroku No 988, 1997. (Kyoto 1996).
[Ued98] Tetsuo Ueda. Critical orbits of holomorphic maps on projective
spaces. J. Geometric Analysis, 8(2):319–334, 1998.
[Wei97] Charles A. Weibel. An Introduction to Homological Algebra. Cam-
bridge University Press, 1997.
	Introduction
	Cohomomorphisms
	Cohomomorphisms and .
	Invariant Global Sections
	Regularity and Positivity
	Subsheaf Cohomology
	Invariant Currents
	Nimble Forms and Lenient Currents
	Hölder Lemmas
	Eigencurrents for Cohomologically Expanding Smooth Maps
	Eigencurrents for Smooth Covering Maps
	Holomorphic Endomorphisms
	ddc Cohomology
	Result via Invariant Sections
	Sections version with an Invariant Ample Bundle
	Bibliography
