AN INEQUALITY BETWEEN ENTROPIES

OF PSEUDOGROUPS OF HOLOMORPHIC GERMS

Xavier Gómez-Mont and Bruno Wirtz

ABSTRACT : For Axiom A diffeomorphisms on compact manifolds, the entropy of fixed points coincides with the topological entropy (see Mañe [5] and Katok and Hasselblatt [4]). Using Ghys, Langevin and Walzcak [2], one extends the definitions of entropies to pseudogroups of germs of biholomorphisms Bih_{0,ℂ}. The topological entropy of these pseudogroups has been estimated in [6] and there exists fixed points for these pseudo-groups [3]. In this paper we show that the entropy of fixed points is greater than or equal to the topological entropy for pseudogroups of germs of biholomorphisms Bih_{0,ℂ} restricted to small rings around 0.

The principle underlying our proof of the inequalities of the entropies is a construction that produces fixed points of a conformal pseudogroup P. We say that x and y are strictly (n, ε) separated if for all words h of the pseudogroup of length less than n and which are defined for x and for y we have that dist(h(x), h(y)) < ε, but there is a word f in the pseudogroup of length n with dist(f(x), f(y)) ≥ ε. These words are in some sense ‘most expanding’. We will find for these words an exponentially small (with respect to the length of the word f) disc U_{f} where the expandig words f are defined, and we will show that f is almost a linear function on U_{f}, in the sense that all the derivatives and all the secants have almost the same value: For any δ′ there is an ϵ_{0} such that if ϵ < ϵ_{0}

| \frac{f′(x)}{f′(y)} − 1| < δ′ , | \frac{f(x) − f(y)}{f′(x)(x − y)} − 1| < δ′ , x, y ∈ U_{f} .  (1)

The strength this result is that the relation ε_{0}(δ′) is independent of the length n of the word f of the pseudogroup, as long as the word is ‘most expanding’. Positive topological entropy of the pseudogroup P produces an exponentially increasing number of such elements of the pseudogroup. This then means that if ε is small, we may find exponentially small discs, which are transformed almost linearly by transformations in the pseudogroup P to size ϵ.

If the pseudogroup P besides having positive topological entropy it has the property of having \frac{ε}{2} -dense orbits, so that we can find a word g of length bounded by n_{0} that transforms the exponentially small disc U_{f} into a subset of f(U_{f}), which is almost a disc of radius ε. By composition, this produces a ‘Brouwer fixed point configuration’: f^{−1} ∘ g(U_{f}) ⊂ U_{f}, and hence a fixed point of the pseudogroup within the exponentially small disc U_{f} by a word of length n + n_{0}.

The property of ‘almost linearity’ (1) of the ‘very expansive’ elements of the pseudogroup is concentrated in what we have called ‘the Dog’s Lemma’ (Lemma 7). If you have a dog on a leash, the leash should be ‘tight’, for otherwise it is useless. The Dog’s Lemma says that if ε is small enough, then the image of the segment [x, y] between two strictly (n, ε) separated points by any partial word of the most expanding words is ‘tight’, that is, the rope between x and y is all the time close to a straight line. Further, there is a smaller disc around x such that for any end point z in this disc the segment [x, z] will also be ‘tight’ with respect to the word separating x and y. The exponential estimate of the distance between two strictly (n, ϵ) separated points is the chore of the proof of the Dog’s Lemma.

The conformal pseudogroups we use correspond to the restriction to a small ring R_{β}^{α} := {β ≤ |z| ≤ α} the pseudogroup generated by a finite number of germs H_{1} = {g_{1}, …, g_{r}} ⊂ Bih_{0}(ℂ), where the topological entropy and density hypothesis are satisfied generically.

Our main result is:

Theorem 1 : Let H_{1} ⊂ Bih_{0,ℂ} generate a non-linearizable group with dense linear parts in ℂ^{∗}, and denote by H the pseudogroup generated by H_{1} in the ring R_{β}^{α}, with α small. Then there exists a β_{0}, depending on α and H_{1}, such that for any β ∈< 0, β_{0} >, the entropy h_{fix}(H, H_{1}, α, β) of fixed points is greater than or equal to the topological entropy h_{top}(H, H_{1}, α, β).

1. Entropies of Conformal Pseudogroups on Compact Rings.

In this section we fix the hypothesis and notation for the rest of the paper. Let k ≥ 2 be an integer and g_{1}, g_{1}, ... , g_{k} be k germs of holomorphic diffeomorphisms in Bih_{0,ℂ}

g_{j}(z) = λ_{j}z + ..., λ_{j} ≠ 0 , j = 1, 2, ..., k.

We will denote the finite set

H_{1} := {Id} ∪ {g_{j}^{±1}, j = 1, 2, ..., k} := {g_{j}, j = 1, 2, ..., k}.

We suppose moreover that the derivatives g′_{j}(0) generate a multiplicative group dense in ℂ^{∗}. It is a generic condition because outside a subset of ℂ^{2} with null Lebesgue measure, the multiplicative group {λ_{1}^{n}λ_{2}^{m}, (n, m) ∈ ℤ^{2}} is dense in ℂ^{∗}. Therefore one of the germs g_{j}, for example g_{1}, is hyperbolic in a neighbourhood of 0. Hence this germ is linearizable. Let z be its coordinate of linearization:

g_{1}(z) = λ_{1}z.

As the commutators g_{j}^{−1} ∘ g_{k}^{−1} ∘ g_{j} ∘ g_{k} = z + … are tangent to the identity, we have that either all of them are the identity and the group is commutative, or there exists a couple of indices (i, j) such that

g_{i}^{−1} ∘ g_{j}^{−1} ∘ g_{i} ∘ g_{j}(z) = z + μz^{d+1} + ..., μ ≠ 0, d minimal.

Since g_{1} is linear, applying the above formula, we obtain that the other generator germs g_{2}, ... , g_{k} ∈ H_{1} are at least d-tangent to their linear part :

g_{j}(z) = λ_{j}z + μ_{j}z^{d+1} + ..., j = 2, ..., k.

where one at least of the coefficient μ_{j} is not equal to 0. The number d = 1, 2, …, ∞ is infinite if and only all the g_{j} are simultaneously linear. It is an invariant of H_{1} and any element of the group generated by H_{1} in Bih_{0}ℂ may be written in the form

g := g_{i_{s}} ∘ … ∘ g_{i_{1}} = λz + μz^{d+1} + … λ ∈ ℂ^{∗} , μ ∈ ℂ.

Let us fix a real number α, which is positive and strictly inferior to the radius of convergence of the germs in H_{1}. Denote the compact ring by

R_{β}^{α} := {β ≤| z |≤ α} := Δ̄(α) − Δ(β) ⊂ ℂ  ,  Δ(α) := Δ(0, α) := {| z |< α ⊂ ℂ}

and let M be:

M := max{g′_{j}(z) / z ∈ Δ̄(α) , j = 1, …, k}.  (1.1)

Denote by W ord(H_{1}) the free monoid generated by H_{1}, its elements correspond to words formed with the alphabet H_{1}, and where there is no cancellations. The pseudogroup H := H_{β}^{α}(H_{1}) generated by H_{1} on the ring R_{β}^{α} consists of pairs (f, Dom(f)) formed by a word f ∈ W ord(H_{1}) together with its domain of definition: If f := f_{n} ∘ f_{n−1} ∘ ... ∘ f_{1} ∈ W ord(H_{1}) then

Dom(f) := {z ∈ R_{β}^{α} / f_{k} ∘ f_{k−1} ∘ ... ∘ f_{1}(z) ∈ R_{β}^{α}, k = 1, 2, ..., n}.

It is possible that Dom(f) = ∅ for some word f. For example, the domain of g_{1}^{n} is empty if | λ_{1} |^{n}≥ α/β.

The length of a word is the number of letters it has, and we will denote by H_{n} ⊂ H the collection of words of length n. Let f := f_{n} ∘ ... ∘ f_{1} be a word of length n such that Dom(f) is not empty, and x ∈ Dom(f). The partial images of x by f are defined as

x_{0} = x, x_{1} = f_{1}(x_{0}), x_{2} = f_{2}(x_{1}), ... x_{n} = f_{n}(x_{n−1}).

A fixed point of length n of the pseudogroup H is a point x of the ring R_{β}^{α} ⊂ ℂ such that there

∃(f, Dom(f)) ∈ H_{n} such that x ∈ Dom(f) , f(x) = x , f|_{V_{x}} ≠ id_{V_{x}}

where V_{x} is a neighbourhood of x. Let F ix(H, H_{1}, n, α, β) be the set of all fixed points of words of length n and denote by the symbol # the cardinality of a finite set. The entropy of fixed points of the pseudo-group H relatively to the generator set H_{1} and restricted to the compact ring R_{β}^{α} is

h_{fix}(H, H_{1}, α, β) := lim sup_{n→+∞} \frac{1}{n} ln(#(F ix(H, H_{1}, n, α, β))).

Let ε be a strictly positive real number and n a non negative integer. The points x, y ∈ R_{β}^{α} are (α, β, n, ε) separated by the pseudo-group H if and only if

∃(f, Dom(f)) ∈ H_{n} such that x, y ∈ Dom(f) and | f(x) − f(y) | ≥ ε.

A subset of R_{β}^{α} is (α, β, n, ε) separated by the pseudo-group H if and only if any two distinct elements of this subset are (α, β, n, ε) separated.

The topological entropy of the pseudo-group H relatively to the generator set H_{1} and restricted to the compact ring R_{β}^{α} is

h_{top}(H, H_{1}, α, β) := lim_{ε→0} lim sup_{n→+∞} \frac{1}{n} ln(max\{#(A), A ⊂ R_{β}^{α} is (α, β, n, ε) − separated\}).

2. Strictly Separating Points and its Separating Words

Points x, y ∈ R_{β}^{α} which are (α, β, n, ε) separated but not (α, β, n − 1, ε) separated by the pseudogruop generated by H_{1} will be called strictly (α, β, n, ε) separated points, and the word f = f_{n} ∘ f_{n−1} ∘ ... ∘ f_{1} will be called a separating word. We will use the notation

ϕ_{m} := f_{m} ∘ ... ∘ f_{1}  ,  m = 1, …, n

for the partial words of the separating word f.

Proposition 2 : There exists constants K_{1}, K_{2} ∈< 0, ∞ >, K_{1} depending on H_{1} and K_{2} depending on α, β and H_{1}, such that for any strictly (α, β, n, ε) separating points x and y we have:

| x − y |< εK_{2}e^{−nK_{1}α^{d}}  (2.1)

Proof : The strictly positive lower bound of the topological entropy of the pseudo-group shown in [6] implies directly the Proposition. □

Let x_{0}, x_{1}, ..., x_{n} and y_{0}, y_{1}, ..., y_{n} be the partial images by the word f = f_{n} ∘ f_{n−1} ∘ ... ∘ f_{1} strictly separating x and y..

Proposition 3. For every δ ∈< 0, 1 > there exists α_{1} > 0 such that for every strictly (α, β, n, ε) separating points x and y with α < α_{1} we have

∑_{0≤j≤n} | x_{j} − y_{j} | < \frac{K_{2}}{K_{1}α^{d}δ} ε.  (2.2)

Proof. As x_{j} and y_{j} are strictly (n − j, ε) separated, from (2.1) we obtain :

| x_{j} − y_{j} | ≤ εK_{2}e^{(−n+j)K_{1}α^{d}}.  (2.3)

It follows then that

∑_{0≤j≤n} | x_{j} − y_{j} | ≤ εK_{2} ∑_{0≤j≤n} e^{(−n+j)K_{1}α^{d}} =

= εK_{2} \frac{1 − e^{(−n−1)K_{1}α^{d}}}{1 − e^{−K_{1}α^{d}}} < εK_{2} \frac{1}{1 − e^{−K_{1}α^{d}}}  (2.4)

We have that 1 + t ≤ e^{t} and the tangent at the point (0, 1) of the graph of e^{t} is the graph of 1 + t. For any δ ∈< 0, 1 > the graph of the affine function 1 + δt is above the graph of e^{t} for an interval < −α_{0}, 0 >. Hence for t ∈< −α_{0}, 0 > we have

1 + t < e^{t} < 1 + δt ⟹ t < e^{t} − 1 < δt ⟹ \frac{1}{δt} < \frac{1}{e^{t} − 1} < \frac{1}{t}

Setting t = −s we obtain for s ∈< 0, α_{0} >

− \frac{1}{δs} < \frac{1}{e^{−s} − 1} < − \frac{1}{s} ⟹ \frac{1}{s} < \frac{1}{1 − e^{−s}} < \frac{1}{δs}

Applying this to (2.4) we obtain (2.2) with α_{1} = ( \frac{α_{0}}{K_{1}} )^{\frac{1}{d}}. Clearly δ tends to 1 if and only if α_{1} tends to 0. □

Lemma 4. For any g_{j} ∈ H_{1} there is a holomorphic function

G_{j} : Δ(α) × Δ(α) → ℂ  \frac{g′_{j}(x)}{g′_{j}(y)} = e^{(x−y)G_{j} (x,y)}  (2.5)

with G_{j} having in its power series expansion only terms of degree at least d − 1. There is a constant K_{3} > 0 such that

|G_{j}(x, y)| < K_{3}|(x, y)|^{d−1}

for all j and |x| < α_{0} , |y| < α_{0}.

Proof. We have a power series expansion g(z) = a_{1}z + ∑_{j≥d+1} a_{j}z^{j}, then

g′(x) − g′(y) = ∑_{j≥d+1} ja_{j}(x^{j−1} − y^{j−1}) = (x − y) ∑_{j≥d+1} ja_{j}(∑_{ℓ=0}^{j−2} x^{j−2−ℓ}y^{ℓ})

and defining

L(X, Y) := \frac{g′(x) − g′(y)}{g′(y)(x − y)} = [ ∑_{j≥d+1} ja_{j}(∑_{ℓ=0}^{j−2} x^{j−2−ℓ}y^{ℓ})]( \frac{1}{a_{1}} − \frac{(d + 1)a_{d+1}}{a_{1}^{2}} y^{d} + …) = O(x, y)^{d−1}

we have

\frac{g′(x)}{g′(y)} = 1 + (x − y)L(x, y)

The principal value of its logarithm is then

log[ \frac{g′(x)}{g′(y)} ] = (x − y)L(x, y) − \frac{(x − y)^{2}L(x, y)^{2}}{2} + \frac{(x − y)^{3}L(x, y)^{3}}{3} − …:= (x − y)G(x, y)

G(x, y) := L(x, y) − \frac{(x − y)L(x, y)^{2}}{2} + \frac{(x − y)^{2}L(x, y)^{3}}{3} − … = O(x, y)^{d−1}

from which we deduce the first part of the Lemma, on using g_{j} for g.

We may write

G(x, y) = ∑_{j=0}^{d−1} a_{j}(x, y)x^{d−1−j}y^{j}

from where we deduce

\frac{|G|}{|(x, y)|^{d−1}} ≤ ∑_{j=0}^{d−1} |a_{j}(x, y)| < K_{3}

Since there are only a finite number of elements in H_{1} we deduce from this the second part of the statement. □

We now show that the rate f′(x)/f′(y) is O(ε) close to 1:

Proposition 5. For every δ′ > 0 there exists ϵ_{0} > 0 such that for every ε < ϵ_{0} and for every strictly (α, β, n, ε) separating points x, y ∈ R_{β}^{α} with separating word f, we have

| \frac{f′(x)}{f′(y)} − 1| < δ′  (2.6)

Remark : The previous statement has to be understood as a result in ℂ, not only a result about the modulus of f′(x)/f′(y)

Proof. If the word f = f_{n} ∘ f_{n−1} ∘ ... ∘ f_{1} we have by the chain rule and (2.5)

\frac{f′(x)}{f′(y)} = \frac{f′_{n}(x_{n−1})}{f′_{n}(y_{n−1})} … \frac{f′_{1}(x_{0})}{f′_{1}(y_{0})} = e^{(x_{n−1}−y_{n−1})G_{n}(x_{n−1},y_{n−1})} … e^{(x_{0}−y_{0})G_{1}(x_{0},y_{0})} =

= e^{(x_{n−1}−y_{n−1})G_{n}(x_{n−1},y_{n−1})+...+(x_{0}−y_{0})G_{1}(x_{0},y_{0})}

By Proposition 3 and Lemma 4 we have that

|(x_{n−1} − y_{n−1})G_{n}(x_{n−1}, y_{n−1}) + … + (x_{0} − y_{0})G_{1}(x_{0}, y_{0})| < \frac{K_{2}K_{3}α^{d−1}}{K_{1}α^{d}δ} ε = \frac{K_{2}K_{3}}{K_{1}αδ} ε

The exponential function restricted to a disc Δ̄(r) of radius r is most expanding at the point z = r (since \frac{d}{dz} e^{z} = e^{z} and r ∈ Δ̄(r) is the point with the largest real part), hence e(Δ̄(r)) is contained in the disc of radius re^{r}. We may then choose

ϵ_{0} := min\{1, δ′ \frac{K_{1}αδ}{K_{2}K_{3}} e^{− \frac{K2K3}{K1αδ}} \}

□

We now show that the ratio \frac{f(x)−f(y)}{f′(x)(x−y)} is O(ε)-close to 1:

Proposition 6. For every δ′ > 0 there exists ϵ_{0} > 0 such that for every ε < ϵ_{0} and for every strictly (α, β, n, ε) separating points x, y ∈ R_{β}^{α} with separating word f, we have

| \frac{f(x) − f(y)}{f′(x)(x − y)} − 1| < δ′  (2.7)

Proof. For any element g_{j} ∈ H_{1} we have that the holomorphic function g_{j}(x) − g_{j}(y) on Δ(α) × Δ(α) vanishes along the diagonal x − y = 0, so we may divide to obtain a holomorphic function \frac{g_{j} (x)−g_{j} (y)}{x−y} on Δ(α) × Δ(α). Its restriction to the diagonal is g′_{j}. Hence there is a holomorphic function L_{j}(x, y) on Δ(α) × Δ(α) whose power series expansion around 0 has no constant term, such that

\frac{g_{j}(x) − g_{j}(y)}{g′(y)(x − y)} = 1 + … = e^{(x−y)L_{j} (x,y)}

Hence we may find a constant K_{4} such that

|L_{j}(x, y)| < K_{4}  for x, y ∈ Δ(α  ,  j = 1, …

The quotient in the statement of the Proposition may be decomposed as a product:

\frac{f(x) − f(y)}{f′(x)(x − y)} = ∏_{0≤j≤n−1} \frac{x_{j+1} − y_{j+1}}{f′_{j+1}(x_{j})(x_{j} − y_{j})} = e^{(x_{n−1}−y_{n−1})L_{n}(x_{n−1},y_{n−1})+...+(x_{0}−y_{0})L_{n}(x_{0},y_{0}))}.

Again we have by Proposition 3

|(x_{n−1}−y_{n−1})L_{n}(x_{n−1}, y_{n−1}))+…+(x_{0}−y_{0})L_{n}(x_{0}, y_{0})| < K_{4}(|x_{n−1}−y_{n−1}|+…+|x_{0}−y_{0}|) < \frac{K_{2}K_{4}}{K_{1}α^{d}δ} ε.

We may similarly as in Proposition 5 choose

ϵ_{0} := min\{1, δ′ \frac{K_{1}α^{d}δ}{K_{2}K_{4}} e^{−_{\frac{K2K4}{K1αdδ}}} \}

□

3. The Dog’s Lemma and The Almost Linear Separating Words

Define the distortion of a compact curve as the division of its length by the distance between its extremities. It is always larger than 1, and equal to 1 if and only if the curve is a linear segment. The distortion is invariant by a linear conformal map.

Lemma 7 (The Dog’s Lemma). Let α be small, and for ε small enough relatively to H_{1} and β and f a separating word for the strictly (α, β, n, ε) separating points x and y, we have

1) The domain of definition of the word f on the ring R_{β−ε}^{α} contains the line segment [x, y] and the distortion of the curve f([x, y]) is O(ε)-close to 1.

2) The domain of definition of the word f on the ring R_{β−ε}^{α+ε} contains the disc Δ(x, r|x − y|) and the distortion of the curve f([x, z]) is O(ε)-close to 1 for any z ∈ Δ(x, r|x − y|), where r := \frac{1−δ′}{(1+δ′)M^{2}} .

Proof. We begin by choosing α small enough so that Propositions 3 holds, and for a sufficiently small δ′ let ϵ be small enough so that Propositions 4 and 5 are satisfied, and further assume the generator maps g_{j} are a small perturbation of their linear parts on the disc Δ(α). Let ε << β be small enough so that the elements in H_{1} have an analytic continuation to the disc Δ(α + ε).

If f = f_{n} ∘… f_{1} is a separating word for x and y, we show by induction on m that for z ∈ [x, y] the partial words ϕ_{m}(z) := f_{m} ∘ … f_{1}(z) is well defined in the larger ring R_{β−ε}^{α} and that the function

[x, y] → ℝ  ,  z → |ϕ_{m}(z) − ϕ_{m}(x)|  (3.1)

is an increasing function of z ∈ [x, y]. Note that it is a consequence of the arguments provided is section 2 that the equation (2.2) for z ∈ [x, y] instead of y implies that (2.6) and (2.7) are valid for z:

∑_{0≤j≤n} | x_{j} − z_{j} | < \frac{K_{2}}{K_{1}α^{d}δ} ε  ⟹  | \frac{f′(x)}{f′(z)} − 1| < δ′ , | \frac{f(x) − f(z)}{f′(x)(x − z)} − 1| < δ′
(3.2)

Suppose by induction that the hypothesis (3.1) is true for j = 1, … m − 1 and that it fails for j = m. We first note that the hypothesis of (3.2) is true for ϕ_{m−1} and z ∈ [x, y]:

∑_{0≤j≤m−1} | x_{j} − z_{j} |< ∑_{0≤j≤m−1} | x_{j} − y_{j} |< \frac{K_{2}}{K_{1}α^{d}δ} ε

and hence we also have

| \frac{ϕ′_{m−1}(x)}{ϕ′_{m−1}(z)} − 1| < δ′  | \frac{ϕ_{m−1}(x) − ϕ_{m−1}(z)}{ϕ′_{m−1}(x)(x − z)} − 1| < δ′  (3.3)

Let z ∈< x, y > be an interior point of the line segment [x, y] where the function (3.1) attains a local maximum (at this point the leash is not tight). At z_{m} := ϕ_{m}(z) the tangent vector ϕ′_{m}(z)(z − x) of the curve ϕ_{m}([x, y]) is tangent to the circle Δ(x_{m}, |z_{m} − x_{m}|) (since it is an extremum point of the distance function (3.1)), which is ℝ-proportional to i(z_{m} − x_{m}):

ϕ′_{m}(z)(z − x) = is(z_{m} − x_{m})  s ∈ ℝ.  (3.4)

But from (3.3)

ϕ′_{m−1}(z)(z − x) ≃ z_{m−1} − x_{m−1}

and applying to the right hand of this expression the linear map f′_{m}(z_{m−1}), and to the left hand side we apply f_{m} which by assumption almost coincide since f_{m} ∈ H_{1} is a small perturbation of its linear part, we obtain

ϕ′_{m}(z)(z − x) ≃ z_{m} − x_{m}

which is a contradiction to (3.4) since 1 is never near a pure imaginary number ir. This contradiction proves that no such local maximum z ∈ [x, y] exists and so (3.2) is true for all partial words and z ∈ [x, y]. This in its turn implies then that (3.3) is satisfied for all partial words ϕ_{m} on [x, y], including f = ϕ_{n}.

A consequence of the non separation of x and z is by (3.3) that the rate f′(x)/f′(z) is O(ε)- close to 1. It implies that the argument of f′(z) is almost constant, O(ε)-close to the argument of f′(x). The length of the curve f([x, y]) is the integral

∫_{<x,y>} | f′(z) | dz.

As the argument of f′(z) is almost constant, the contribution to the integral the change of the argument in the derivative induces only a small error. The length of the curve is O(ε)-close to the distance between its extremities. The distortion of f(< x, y >) is O(ε)-close to 1.

In the above argument, we have not paid attention to the behaviour when the points x_{m} and y_{m} approach the boundary of the ring R_{β}^{α}. Using the description just given, one can see that if x_{m} is approaching the exterior boundary of the ring R_{β}^{α} all the segment [x_{m}, y_{m}] is within the ring, since the exterior boundary is convex, and ϕ_{m}([x, y]) is almost a linear segmet by (3.2). If x_{m} is approaching the interior boundary, then if ε is small enough compared to β, the images ϕ_{m}([x, y]) are very close to the intervals [x_{m}, y_{m}], and so the domain of definition of the word ϕ_{m} in the slightly larger ring R_{β−ε}^{α} contains all the interval [x, y]. This proves part 1.

To prove part 2, we denote Δ_{1} := Δ(x, r|x − y|), and we begin by showing that

z ∈ Δ_{1} ⟹ |z_{m} − x_{m}| ≤ |y_{m} − x_{m}|,  (3.5)

where r is defined in the statement of the Lemma and M in (1.1). We prove (3.5) by induction on the length of the partial word. Suppose it is true for partial words of length less than m, and that it is not true for ϕ_{m}. Summing up the right hand inequalities (3.5) obtained for 1, …, m − 1 we obtain via (3.2) for n = m − 1, that (3.3) is verified on Δ_{1}. The control on the derivatives of ϕ_{m−1} gives

|ϕ′_{m−1}(x)| |x − y|(1 − δ′) ≤ |x_{m−1} − y_{m−1}| ≤ |ϕ′_{m−1}(x)| |x − y|(1 + δ′)
|ϕ′_{m−1}(x)| |x − z|(1 − δ′) ≤ |x_{m−1} − z_{m−1}| ≤ |ϕ′_{m−1}(x)| |x − z|(1 + δ′)

and hence

|x_{m−1} − z_{m−1}| ≤ \frac{1 + δ′}{1 − δ′} \frac{|x − z|}{|x − y|} |x_{m−1} − y_{m−1}| ≤ \frac{|x_{m−1} − y_{m−1}|}{M^{2}}

Apply now f_{m} to this expression, which expands at most by M, to obtain

|x_{m} − z_{m}| ≤ M|x_{m−1} − z_{m−1}| ≤ \frac{|x_{m−1} − y_{m−1}|}{M} ≤ |x_{m} − y_{m}|,

which is a contradiction to the assumption that (3.5) was not true for m ≤ n and some point z ∈ Δ_{1}. Hence (3.5) is true for all the partial words, including f on all Δ_{1}, wherever it is defined. Proceeding now as in part 1, one shows that the word f considered in the pseudogroup generated by H_{1} in the larger ring R_{β−ϵ}^{α+ϵ} is defined on the disc Δ_{1}, and f restricted to Δ_{1} is almost a linear map, satisfying (3.3). This finishes the proof of part 2. □

4. Construction of Fixed Points

Using the density of the pseudo-orbits [1] it is possible to come back arbitrarly close to any initial point using another word:

Lemma 9. Given ε > 0 there exists an integer N_{ε}, depending only on ε and H_{1} such that for any x and y in the compact ring R_{β}^{α} there exists a word h of length smaller than N_{ε} such that x is in Dom(h) and h(x) is ε-close to y.

Proof. The statement is obvious if all the germs in H_{1} are linear. In the non-commutative case, assume that x and y are points close to the external circle of radius α. If they are not, then some images of x and y by powers of the hyperbolic homothecy g_{1} are close to this circle. As β is sufficiently small, the homothecy of rate y/x can be approached by conjugacy of some element of H by a contracting power of g_{1}. The existence of N_{ε} is deduced from this construction. □

Proof of Theorem 1. Let ε be small enough, and A a maximal (α, β, ε, n) separated set. Fix x in A and let y in A − {x} be such that | x − y | is minimal. Proposition 2 implies that the distance | x−y | is dominated by an exponentially small estimate. Moreover, Proposition 2 means that there exist j ≤ n such that x and y are strictly (α, β, j, ε) separated. The fraction j/n is minorated by some strictly positive constant. This is also a consequence of Proposition 2. Let f be a word of length j strictly separating x and y. Let Δ_{1} := Δ(x, r|x−y|) be the disc in the Dog’s Lemma. The derivative of f evaluated at Δ_{1} is exponentially large. Let g ∈ H_{N} be a word of length N := N_{\frac{rε}{2}} (from Lemma 9) such that it is defined in the exponentially small disc Δ_{1} and |g(x) − f(x)| < \frac{rε}{2} . Since g has bounded derivative and Δ_{1} is exponentially small, we have that g(Δ_{1}) ⊂ f(Δ_{1}) ≃ Δ(f(x), rε), so that f^{−1} ∘ g : Δ_{1} → Δ_{1} is well defined. By Brouwer’s Fixed Point Theorem, there is a fixed point within Δ_{1}.

Given points x and y which are strictly (α, β, n, ε) separated, two fixed points are exhibited with the previous construction. One is associated to x, the other to y. But these two points could perhaps be the same. The proof is completed by the following technical astuce.

Denote by E the function integral part. We fix n large and ε > 0. Given A a maximal (α, β, n, ε) separated set, given x an element of A we fix y such that | x − y | is minimal in A − {x}. The integer n is supposed large enough so that x and y are not strictly (α, β, n + E(\sqrt{n}), ε) separated. Let f be a germ of H_{n}(α, β) separating x and y. Let x̃_{0} be such that x_{n} and x̃_{0} are strictly (α, β, E(\sqrt{n}), ε) separated. Assume the same conditions holds for y_{n} and ỹ_{0}. Let F be a germ of H_{E(n)} separating x_{n} and x̃_{0}. Let G a germ of bounded length such that G(x) is rε/2-close to F (x_{n}). There exists a fixed point close to x associated to the word f^{−1} ∘F^{−1} ∘G. It is exactly the previous construction. The same construction is also valid for y. Another fixed point is exhibited. These two fixed point are necessarly different because their image by f is close to f(x) for one, close to f(y) for the other, which are ε separated. The order of these two fixed points is smaller than n + E(\sqrt{n}) + N_{\frac{rε}{2}} . Hence the cardinal of fixed point of length n + E(\sqrt{n}) + N_{rε} is larger than the maximal cardinal of (α, β, n, ε) separated points. □

References

1. D. Cerveau, Densite de feuilles de certaines èquations de Pfaff a en variables, Ann. Inst. Fourier (1983), 185-194.

2. E. Ghys, R. Langevin, P. Walzac, Entropie géométrique des feuilletages, Acta Mathematica (1988), 105-142.

3. X. Gómez-Mont, B. Wirtz, On Fixed Points of Conformal Pseudogroups, Bol. Soc. Bras. Mat. (1995), 201-209.

4. A. Katok, B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems, Cambridge U.P. (1995).

5. R. Mañé, Ergodic Theory and Differentiable Dynamics, Springer (1987).

6. B. Wirtz, Entropies de Feulletages Holomorphes de ℂ^{2} à Singularité isolée non Dicritique, Ann. Inst. Fourier, to appear.

Xavier Gómez-Mont CIMAT A.P. 402 Guanajuato, 36000, Mexico gmont@ fractal.cimat.mx Bruno Wirtz Université de Bretagne Occidental wirtz@u-brest.fr