Inverse Limits in Holomorphic Dynamics.

Local Topology and Invariants.

Avraham Goldstein

The City University of NY

Parts of this work has been published in “Topology and Its Applications” journal. Volume 171 (July 2014) pp. 15-34. Co-authored with Carlos Cabrera and Chokri Cherif.

These parts are also available online at arXiv.org. E-print identifier 1202.2900.

I would like to thank Dr. Carlos Cabrera from The National University of Mexico and Dr. Chokri Cherif from BMCC/CUNY for their contributions.

Abstract:

An inverse dynamical system is a sequence

S=S1 ← S2 ← S3 …

of Riemann surfaces Si and branched covering maps fi:Si+1→ Si.

f1 f2

In this work:

•All the Riemann surfaces Si are just copies of the Riemann surface S0 , which is either the unit disk, the complex plane or the Riemann sphere. •All the maps fi are just copies of a branched covering map f0 of some finite degree d>1.

We define the Plaque Inverse Limit [P.I.L.] S∞ of S to be the set of all sequences of points

x = ( x1 S∈ 1, x2 S∈ 2, … | fi(xi+1) = xi )

equipped with the topology, in which the open sets are all the sequences of open sets

U = ( U1 S⊂ 1, U2 S⊂ 2, … | fi(Ui+1) = Ui )

and with continuous projections maps pi: S∞ → Si such that

fi p∘ i+1 = pi : S∞ → Si

Constructing an inverse limit of f0:S0 → S0 where degree of f0 is 2.Every S_i here is a copy of S0

An open neighborhood (U_1, U_2, U_3, …) in the Plaque Inverse Limit.Here U_3 contains a critical point c_3 and U_2 contains its image f_2(c_3).

The standard inverse limit of an inverse dynamical system is defined in the similar way, except that in its topology the open sets are all the sequences of open sets

U = ( U1 S⊂ 1, U2 S⊂ 2, … | fi(Ui+1) = Ui and fj-1(Uj) = Uj+1 )

for all j ≥ m, where m is a fixed integer number which depends on U.

Actually, the inverse limit set is, naturally, a subset of the Cartesian product of the underlying sets of Si for i=1,2,….

Thus, it is natural to study:

1.The standard inverse limit, which is a subspace of the direct product of the Si, for i=1,2,…, equipped with the Tychonoff topology;

2.The inverse limit set, equipped with the box topology, which is a subset of the Cartesian product, equipped with the box topology.

P.I.L. has more open sets than the standard inverse limit.

P.I.L. has less open sets than the inverse limit set, equipped with the box topology – for a given U1 one cannot take “arbitrary small” Un.

There is a trivial projection map from P.I.L. onto the standard inverse limit.

It is important to notice, that the map f0:S0→S0 inducees an automorphism f : S∞ → S∞ of the P.I.L. and an automorphism of the standard inverse limit.

We take a different approach: P.I.L. is the inverse limit of a dynamical system in the Category of locally connected topological spaces and continuous open maps.

P.I.L. is a subobject of the direct product of the Si, for i=1,2,…, in that Category. However, P.I.L. is not a subspace of the direct product!

The local base for topology of S∞ at a point x consists of all open sets U containing x, such that:

• Each Ui is conformally equivalent to the unit disk;

• Each fi, restricted to Ui+1, is conformally equivalent to a polynomial map of a degree between 1 and d.

Such open sets U of S∞ are called plaques.

When we speak of a neighborhood of a point in S∞, we assume it to be plaque.

When we speak of a neighborhood of a point in a Riemann surface, we assume it to be simply connected.

A point x in S∞ is called regular if exists a neighborhood U of x and an integer n, such that all the sets Un+i, for i=1,2,…, contain no critical points of fn+i-1.

Thus, a point x in S∞ is regular, if for some its neighborhood U and some integer n, all the maps fn+i-1: Un+i → Un+i-1 are conformal equivalences.

The set of all the regular points of S∞ is denoted by ∆.

Otherwise, a point x in S∞ is called irregular.

The set ∆ is, clearly, open. Each one of ∆’s path-connected components, which are called leafs, has a Riemann surface structure.

In this work we:

1. Show that ∆ is not empty. Even in general cases of an inverse systems.

2. Develop and construct a σ-algebraic machinery, which addresses “countable modulo finite” situations.

3. Using that σ-algebraic machinery, obtain and compute a new local invariants of P.I.L., which we call signatures.

4. Show that a point is regular if and only if all its signatures are trivial.

5. Construct irregular points and compute their signatures for various dynamical situations. Correlate signatures with dynamical properties.

6. Show that a point x is irregular if and only if for any “small enough” neighborhood V of x, deleting x from V breaks a path-connected component of V into an uncountable number of path-connected components. So, P.I.L. is not a topological manifold at x.

7. Show that an irregular point x has signatures of a certain type if and only if for any “small enough” neighborhood V of x, deleting some point y ≠ x from V breaks a path-connected component of V into an uncountable number of path-connected components.

In this work we:

Classical examples of inverse limits of dynamical systems are d-adic solenoids.

A d-adic solenoid is the inverse limit of the iterates of the d-fold covering self-map of the unit circle, where d>1.

These d-adic solenoids are compact, metrizable topological spaces that are connected, but neither locally connected nor path connected.

Solenoids were first introduced by L. Vietoris in 1927 for d=2 and in 1930 by D. van Dantzig for an arbitrary d.

An Historical Note:

In the picture below f : S1→S1 is just the doubling map of the unit circle.

The [standard] inverse limit – the dyadic solenoid – is isomorphic to the Cartesian product of the unit circle and the Cantor set.

In 1992 D. Sullivan introduced Riemann surface laminations, which arise when taking inverse limits in complex dynamics. A Riemann surface lamination is locally the product of a complex disk and a Cantor set. In particular, Sullivan associates such lamination to any smooth, expanding self-maps of the unit circle, with the d-fold covering self-maps being examples of such maps.

In 1997 M. Lyubich and Y. Minsky and, in parallel, M. Su introduced three-dimensional laminations associated with dynamics of rational self-maps of the Riemann sphere. This formalized the theory of Riemann surface laminations associated with holomorphic dynamics.

They consider the inverse limit of iterations of a rational self-map of a Riemann sphere. The associated Riemann surface lamination, is, in many cases, just the set of all regular points of the inverse limit.

However, in general, certain modifications are performed, in order to satisfy the requirement, that the conformal structure on the leafs of the Riemann surface lamination is continuous along the fiber of the lamination.

Basic Definitions:

For a dynamical system f0 : S0 → S0 , where S0 is either the unit disk, the complex plane, or the Riemann sphere, and f0 is a branched covering of a finite degree d>1, we define an inverse dynamical system S as:

S = S1 ← S2 ← S3 …

where S0= S1= S2= ... and f0= f1= f2= ... .

f1 f2

We speak of f0 as of a map from every Si to Si-1.

We denote the critical points of f0 by c1, ..., ck and regard them as points of every Si.

The following 3 definitions and the lemma, which follows, apply in the more general case of an inverse system, in which one does not require all the Riemann surfaces Si and all the branched covering maps fi to be the same:

Definition 1: Plaque Inverse Limit [P.I.L.] S∞ of an inverse system S is the set of all sequences of points

x = ( x1 S∈ 1, x2 S∈ 2, … | fi(xi+1) = xi )

equipped with the topology, in which the open sets are all the sequences of open sets

U = ( U1 S⊂ 1, U2 S⊂ 2, … | fi(Ui+1) = Ui )

and equipped with continuous projections maps pi: S∞ → Si such that

fi p∘ i+1 = pi : S∞ → Si

Notice that S∞ , as a topological space, is regular and first-countable.

We call the underlying topological space of S∞ by T∞.

Definition 2: An open set U S⊂ ∞ is called a plaque if:

• Each Ui S⊂ i is conformally equivalent to the unit disk; and

• Each fi : Ui+1 → Ui is conformally equivalent to the self-map zt of the unit disk, with t between 1 and degree(fi).

All the plaques, containing a point x of S∞ , constitute a local base for the topology of S∞ at x.

All our “open neighborhoods” of a point in a Riemann surface are assumed to be simply connected and in a P.I.L. are assumed to be plaques.

Definition 3: A point x of S∞ is called regular if there exists an open neighborhood U of x such that for some positive integer $n$, all fn+i:Un+i+1→ Un+i are bijections.

Otherwise, x is called irregular.

The set of all the regular points of S∞ is denoted by ∆. Clearly, ∆ is an open set.

In the following lemma we require all Si to be a fixed Riemann surface - either the Riemann sphere, the complex plane, or the unit disk.

Lemma 4: If degree(fi+1) ≤ 2 x degree(fi) then the set ∆ is not empty.

Algebraic Machinery to define local invariants of P.I.L.

* Binary operations ∨ and ∧ defined as coordinate-wise operations or and and, respectively;

The set of all infinite binary sequences with:

* Partial order ≤ which is given by: b ≤ a if a b = a. Equivalently b ≤ ∨a if and only if bi ≤ ai for all i;

* the negation operation ¬ which interchanges 0 and 1 in every coordinate;

* the minimal element 0,0,0,… and the maximal element 1,1,1,… ; and

Two binary sequences are called almost equal if they differ only in a finite amount of places.

is a Boolean algebra.

Definition 5: The set I is the Boolean algebra of all classes of almost equalbinary sequences, equipped with the ∨ and ∧ operations, which aredefined as follows:

[a1 , a2 , ...] ∨ [b1 , b2 , ...] = [a1 ∨ b1 , a2 ∨ b2 , ...] and

[a1 , a2 , ...] ∧ [b1 , b2 , ...] = [a1 ∧ b1,a2 ∧ b2 , ...].

Its minimal element is 0 = [0 , 0 , ...] and

its maximal element is 1 = [1 , 1 , ...].

Its negation is: ¬ [a1 , a2 , ...] = [¬ a1 , ¬ a2 , ...].

Almost equality is an equivalence relation, which respects the operations , , ¬, the partial order, and the minimal and maximal elements. ∨ ∧

Definition 6: For every element a of I, we define the subset α(a) of I as the set of all b I, such that ∈ b ≤ a.

Note that:α(a) ∪ α(b) ⊆ α(a ∨ b)

α(a b ) = ∧ α(a) ∩ α(b)

α(0) = { 0 }

α(1) = I

Definition 7: The σ-lattice A, spanned by all α(a), where a ∈ I, with the operations and ∩, the minimal element { ∪ 0 }, and the maximal element I, is called the signature σ-lattice. The elements of A are called signatures.

Clearly, (⊂ subset) defines a partial order on A.

This partial order is consistent with the partial order ≤ of I under the map α

Definition 7: For every integer m, the shift map shiftm : I → I takes an equivalence class [i] to the equivalence class of the binary sequence, obtained from i by:

• adjoining m initial 0 entries to the binary sequence i, if m ≥ 0;

• dropping m initial places from the binary sequence i, if m<0.

Notice, that shift0 = IdentityI and shiftm shift∘ -m = IdentityI , since a finite number of entries of a binary sequence is irrelevant for the equivalence class.

Lemma 9: The maps shiftm : I → I induce maps shiftm : A → A .

Again, shift0 = IdentityA and shiftm shift∘ -m = IdentityA .

The following is a crucial property of the image α(I) of I in the σ-lattice A.

Let [i1], [i2], [i3], ... and [t1], [t2], [t3], ... be elements of I.

Theorem 10: If α[i1] ∪ α[i2] ∪ α[i3] ... = ∪ α[t1] ∩ α[t2] ∩ α[t3] ∩ ...

then there exist some natural numbers m and n such that

α[i1] … ∪ ∪ α[im] = α[t1] ∩ … ∩ α[tn]. So, [i1] ∨ ... ∨ [im] = [t1] ... [t∧ ∧ n].

Corollary 11: If α[i1] ∩ α[i2] ∩ α[i3] ∩ ... = α[i] for some [i] in I, then there exists a finite number n such that [i1] ∧ ... ∧ [im] = [i].

Recall, that the projection maps onto Si is a part of the structure of the P. I. L.

Now we define an important local invariant of the P. I. L.

Signature - a local invariant of P.I.L.

Definition 12: For an open neighborhood U in S∞ and a critical point c in S0, the index ind(U,c) I ∈ of U with respect to c is the class of the binary sequence, which has 1 in its nth place if and only if c is contained in Un.

Let T be a regular, first-countable topological space and z be a point in T.

A sequence of open neighborhoods (U(1), U(2), ...) of z shrinks to z if U(i+1) U(i), for all i, and the set {U(1), U(2), ...} is a local basis for the ⊂topology at z.

Clearly, if V U, then ⊂ ind(V,c) ≤ ind(U,c) .

Definition 13: For a point x in S∞ and a critical point c in S0, we define the signature of x with respect to c as:

sign(x,c) = ∩ α( ind(U(j),c) ),

where (U(1), U(2), ...) is an arbitrary sequence of open neighborhoods of x in S∞ shrinking to x.

j=1

∞

Lemma 14: The signature sign(x,c) does not depend on the choice of the sequence (U(1), U(2), ...).

Lemma 15: A point x in S∞ is regular if and only if sign(x,c) = { [0,0,0,...] } for every critical point c.

If sign(x,c) = { [0,0,0,...] }, we say that this signature is trivial.

Lemma 16: For any points x and x′ in S∞ , if sign(x,c) ∩ sign(x′,c) contains any element other than [0,0,0,...], then x = x′.

Lemma 17: For any integer m and any point x in S∞, we have:

sign( f m(x) , c ) = shift-m( sign(x,c) ).

If sign(x,c) = α[i], we say that this signature has a maximal element.

Otherwise, we say that this signature does not have a maximal element.

If sign(x,c) = α[i], where the binary sequence i is the block of k zeros, one, and n-k-1 zeros, repeated periodically, we call this a periodic signature.

Due to Lemmas 16 & 17, this is the only possible case of a nontrivial periodic signature.

Now we investigate some irregular points, which arise is holomorphic dynamics, and compute their signatures.

Signatures and Dynamics.

From the Proposition 2.3 of “The dynamics of rational transforms: the topological picture” (by M. Lyubich) we deduce the existence of examples of maps

fw(z) = 1 +

with a complex parameter w, such that we can start from any point x1 on the Riemann sphere, take some appropriate pre-images of x2, x3, … , and obtain an irregular point x = (x1, x2, ...) in S∞.

Since for any map fw the only pre-image of the critical point ∞ under fw is the critical point 0, the equality

sign (x,0) = shift1( sign (x,∞) )

is satisfied for all points x S∈ ∞ and all functions fw.

wz2

Signature Example 1

End of Example 1

Example 18: Consider the Riemann Sphere S0 and the map f0=zd+c , where c is a “sufficiently small” complex number.

Two points ∞ and ζ, where is ζ the smallest solution of zd - z + c = 0, are the only two attracting fixed points of f0 : S0 → S0.

The entire Riemann Sphere S0 consists of three pairwise disjoint sets:

1.The immediate basin of attraction of ζ.

2.The immediate basin of attraction of ∞.

3.A closed Jordan curve, separating these two basins of attraction.

The points (ζ, ζ, ζ, … ) and (∞, ∞, ∞, …) of S∞ are called the invariant lifts of the fixed points – the cycles of the length one – f0(ζ) = ζ and f0(∞) = ∞, respectively, to S∞. These two points are the only irregular points of S∞.

We will come back to this example in the next section.

Definition 19: Let J S⊂ 0 be such that f(J) = J. Subspace of S∞ , consisting of all points (x1 J, x∈ 2 J, …) S∈ ∈ ∞ , is called the invariant lift of J to S∞ .

A brief review of Holomorphic Dynamics:

A periodic cycle of length n is a set of pairwise different points {z1,...,zn} in S0 , such that f(zi+1) = zi and f(z1) = zn.

The multiplier λ of that cycle is defined as the derivative of f n taken at any one of its fixed points z1, ..., zn. The multiplier is a conjugacy invariant, associated with every periodic cycle.

When | λ | < 1, the cycle is contained in n open neighborhoods, which are cyclically permuted by f. The cycle attracts to it all the points of these neighborhoods. These neighborhoods always contains a critical point of f n.

If λ= 0, the cycle is called super-attracting, otherwise it is called attracting.

When | λ | > 1, the cycle repels from it all the points of some its open neighborhood. Such cycle is called repelling.

1 Rotation number θ = — is rational – the cycle is called parabolic:

When | λ | = 1, so λ = e2πθi, the cycle is called neutral and θ is called the rotation number of the cycle.

There are three different cases of neutral cycles:

pq

Then, locally, at zi , f n(z) = zi + λ(z - zi) + a(z - zi)m +1 + O( (z - zi)m+2 )

where the number m, which is ≥ 1, is called the multiplier of the cycle.

Each point of the cycle is contained in the boundary of m pairwise disjoint open neighborhoods. These n x m neighborhoods are permuted by f. The cycle eventually attracts to it all the points of these neighborhoods. These neighborhoods always contain critical points of f n x m .

The dynamics at a parabolic cycle is described by the Leau-Fatou flower theorem. For example, at a parabolic fixed point with multiplier m there are m rays, with angles 2π / m between them, along which f repels, and m rays, bisecting between the repelling rays, along which f attracts.

The number of different critical points of f , which are contained in these n x m neighborhoods, permuted by f, depends on the dynamical propertied of the parabolic cycle.

2 Rotation number θ is irrational and “badly approximable” by rational numbers – the cycle is called Siegel.

A Siegel cycle is contained in n open neighborhoods, which are cyclically permuted by f. In each one of these neighborhoods, f n is conformally conjugate to a rotation by θ.

3 Rotation number θ is irrational and “well approximable” by rational numbers – the cycle is called Cremer. Locally, at zi , f n is not conjugate toz → zi + λ(z - zi) in any open neighborhood. Also, no open neighborhood has all its points attracted to zi by f n .

For attracting, super-attracting and parabolic cycles, the above-described open neighborhoods are called their immediate basins of attractions.

For Siegel cycles, the above-described open neighborhoods are called their Siegel disks.

- Irrational rotation numbers, up to ±, are topological invariants of f n .

Theorem 20: If x is the invariant lift of:

•A repelling cycle or a Siegel cycle, then x is a regular point. So, sign(x,c) is trivial for every critical point c;

•Either an attracting cycle, a super-attracting cycle, a parabolic cycle, or a Cremer cycle, then x is an irregular point. Thus, by Lemma 15, x must have a non-trivial signature with respect to some critical point. With respect to every critical point c, sign(x,c) = shift ±n( sign(x,c) ).

• In the attracting and the super-attracting cases the signature of x with respect to every critical point c is periodic.

• In the parabolic case the signature of x with respect to some critical point is periodic.

• In the Cremer case the signature of x with respect to some critical point does not have a maximal element.

Signature Example 2

End of Example 2

Lemma 21: If the boundary of the immediate basin of attraction of a parabolic cycle does not contain a critical point, then the invariant lift of that parabolic cycle has only periodic signatures.

Lemma 22: The boundary δB of the immediate basin of attraction of the parabolic fixed point 0 of the cubic map

fa(z) = z + az2 + z3

with a ≠ 0 is a Jordan curve and there exists a homeomorphism φ : δB → S1 which conjugates fa and z → z2 .

Lemma 23: There exist complex numbers a, for which the cubic map fa(z) has a critical point c in δB and c is not pre-periodic, meaning fa

n(c) ≠ fam(c)

for n ≠ m.

Signature Example 3

Let W be an open Siegel disk or an open Herman ring in a complex plane or a Riemann sphere. Let δW be the boundary of W.

Theorem 24: Every point x of the invariant lift of δW is irregular.

Let 2πθ be the angle between the points φ(0) = +1 and φ(c) in S1. Since c is not pre-periodic, θ is irrational and does not depend on the choice of φ.

Let x S∈ ∞ be the invariant lift of the parabolic fixed point 0 of fa .

The signature sign(x,c) is the set of all the classes [a] of binary sequences a, for which there exists a sequence (ε1, ε2, ...) of decreasing positive real numbers converging to 0, so that for any positive integer j, the entries aj, aj+1, aj+2, ... of a, can (but not have to) be 1 only if 2j⋅θ is within εj distance from some integer number mj.

Notice, that the signature in this case does not have a maximal element.End of Example 3

Theorem 25: There exists a Siegel disk W with a rotation number θ, such that its boundary δW is a Jordan curve, which contains a critical point c.

Let x = (x1, x2, ...) be any point of the invariant lift Λ of δW to the P.I.L.

Let ρ : δW → S1 be the unique homeomorphism, such that ρ(c) = +1 and ρ f ∘ ∘ ρ-1 is the clockwise rotation of S1 by the angle 2πθ.

Denote by τ the angle from +1 to ρ(x1), measured counterclockwise, divided by 2π.

The signature sign(x,c) is the set of all the classes [a] of binary sequences a, for which there exists a sequence (ε1, ε2, ...) of decreasing positive real numbers converging to 0, so that for any positive integer j, the entries aj, aj+1, aj+2, ... of a, can (but not have to) be 1 only if τ + θj is within εj distance from some integer number mj.

Notice, that the signature in this case does not have a maximal element.

Signature Example 4

End of Example 4

A quadratic self-map f(z) = z2 + a of the complex plane is called infinitely renormalizable, if there exists an infinite sequence ({U(i), V(i), n(i)}i=1,2,… ) in which:

• Each integer n(i) is a factor in the integer n(i+1);

• The sets V(1),V(2), ... and U(1),U(2), ... are open neighborhoods of 0, satisfying U(i) V(i) and V(i+1) U(i) for all i; and⊂ ⊂

• Each restriction f n(i) : U(i) → V(i) is a branched covering map of degree two.

An infinitely renormalizable map f(z) = z2 + a has a priori bounds, if for some number δ > 0 the neighborhoods U(i) and V(i) can be chosen in such a way, that the modulus of the closed annulus V(i) - U(i) is greater than δ for all i.

A famous example of an infinitely renormalizable f(z) = z2 + a with a priori bounds is the Feigenbaum map with a = -1.401155…. Other such maps can be easily obtained from the Madelbrot set.

The Mandelbrot set – the set of complex numbers a in f(z) = z2 + a for which forward iterations f(0), f 2(0), f 3(0), … do not approach infinity.

Periods of hyperbolic components of the Mandelbrot set.

The map f(z) = z2 + a, with a belonging to these components, has an attracting or a super-attracting cycle of the given period.

Let f be an infinitely renormalizable map with a priori bounds.

Theorem 25: In the plaque inverse limit S∞ of f : ℂ→ there exists ℂexactly one point x = (x1, x2, ...) such that x1= 0 and each xn(i)+1 is a pre-image of x1 under f -n(i) contained inside U(i). This point x is irregular.

The signature sign(x,0) contains a non-zero element β of I, which is the set of all classes [b] of binary sequences b, satisfying the following condition:

For the binary sequence b there exists a sequence ( e(1), e(2), ... ) of increasing positive integers, such that for any integer j > 0, an entry bk of b with k ≥ e(j) can (but does not have to) be 1 only if k is of the form 1+n(t)+n(t+1)+...+n(m), where t and m are any integers satisfying j ≤ t ≤ m.

This element β cannot be described as α[b] for any binary sequence b, which implies that β does not have a maximal element. The signature sign(x,0) also does not have a maximal element.

Signature Example 5

End of Example 5

Let sq = (z(1), z(2), …) be a sequence of points in a regular, first-countable topological space T.

We say that a path p:[0,1] → T passes through sq in the correct way if exist some 0 ≤ t1 ≤ t2 ≤ … ≤ 1 such that z(m) = p( tm ) for all m = 1, 2, … .

Signatures and Local Topology of Plaque Inverse Limits

Lemma 26: There exists a path p:[0,1] → T which passes through sq in the correct way if and only if sq converges to some point z of T and

p( lim (tm) ) = z.m → ∞

The following lemma is a corollary of the fact that there is a continuous bijection from the plaque inverse limit onto the standard inverse limit:

Lemma 28: For every irregular point x S∈ ∞, there exists an openneighborhood U of x, such that for any open neighborhood V U of x, there ⊂are infinitely many positive integers n(1), n(2), ... , for which

Vn(i) contains some critical points of fwhile

(U-V)n(i) does not contain any critical points of f.

The following theorem distinguishes the local topology at the regular andirregular points. In particular, there is no manifold structure at the irregular points.

Lemma 27: Let x = (x1, x2, ...) be a point in S∞ and sq = (z(1), z(2), …) be a sequence of points in S∞ , such that for each positive integer n exists some positive integer m(n), so that z(m)n= xn for all m ≥ m(n). If sq has a converging subsequence in S∞ then the limit of that subsequence is x.

Theorem 29: For every irregular point x S∈ ∞ , there exists some openneighborhood U of x, such that the deleting x from U breaks a path-connected component of U into an uncountable number of path-connected components.

An uncountable number of path-connected components of U – {x} are all “path-connected” to each-other via the irregular point x.

The proof uses Lemma 26

In our Example 18, the regular set of S∞ is a disjoint union of logarithmic Riemann surfaces [infinite spirals]. They all are glued together at their “limit points” (ζ, ζ, ζ, … ) and (∞, ∞, ∞, …) – the only irregular points of S∞.

How different is the local topology at irregular points, which have signatures without maximal elements, from the local topology at irregular points, which do not have such signatures?

Let x S∈ ∞ , be an irregular point, such that sign(x,c), with respect to some critical point c, has no maximal element.

Lemma 30: There exists an open neighborhood U of x, such that for any open neighborhood V U of x, there exists some open neighborhood W of x, with ⊂W V, such that there are infinitely many positive integers n(1), n(2), ... , for ⊂which

(V - W)n(i) contains the critical point c while

(U-V)n(i) does not contain any critical points of f.

Compare with Lemma 28

In the neighborhood U of x: For any neighborhood V of x, we can find some neighborhood W of x, with its closure contained in V, such that an infinite number of levels (V - W)n [open annuli] of (V - W) contain c, while these same levels of (U - V) do not contain any critical points of f.

UV

Lemma 31: The open neighborhood W and the positive integers n(1), n(2), ... in Lemma 30 can be selected in such a way, that the sequence ( f n(1)-1(c), f n(2)-1(c), ... ) converges to some point v1 (V∈ – W)1 .

Lemma 31 plays the role of Lemma 28 for signatures with no maximal elements

Theorem 32: Let x S∈ ∞ be an irregular point, such that for some critical point c, the signature sign(x,c) has no maximal element. There exists some open neighborhood U of x, such that for any open neighborhood V U of x, ⊂there exists a point v V, different from x, such deleting v from V breaks a ∈path-connected component of V into an uncountable number of path-connected components.

The proofs of these two theorems use Lemma 27

Compare with Theorem 29

Theorem 33: Let x S∈ ∞ be an irregular point, such that the signatures sign(x,c) have maximal elements for all critical points c. There exists some open neighborhood U of x, such that for any open neighborhood V U of x ⊂and any point v V, different from x, the open set V - {v} is path-connected.∈

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