Download MINIMAL NUMBER OF PERIODIC POINTS FOR SMOOTH SELF

MINIMAL NUMBER OF PERIODIC POINTS FOR
SMOOTH SELF-MAPS OF S 3
GRZEGORZ GRAFF AND JERZY JEZIERSKI
Abstract. Let f be a continuous self-map of a smooth compact
connected and simply-connected manifold of dimension m ≥ 3, r a
fixed natural number. A topological invariant Drm [f ], introduced
in [5], is equal to the minimal number of r-periodic points for all
smooth maps homotopic to f . In this paper we calculate Dr3 [f ] for
all self-maps of S 3 .
1. Introduction
The classical problem in periodic point theory is to determine or estimate the least number of r-periodic points in homotopy class of a given f ,
a self-map of a compact manifold M m , where r is a fixed natural number
(cf. [10]). If M m is simply-connected and has the dimension m ≥ 3, then
one can always find a map g homotopic to f with only one point in Fix(g r )
(cf. [9]). This is however impossible, if we demand additionally that g is
smooth. In [5] the authors define the topological invariant Drm [f ] equal
to the minimal number of elements in Fix(g r ) for all g which are smooth
and homotopic to f . This enables to explore the new smooth branch of
Nielsen periodic point theory. Let us remark that Drm [f ] may be interpreted purely in the terms of smooth category, namely we may assume
that f is smooth and approximate the homotopy which joins f with g by
a smooth one. Then Drm [f ] gives the minimal number of periodic points
in smooth homotopy class of the given f .
This invariant is obtained by decomposing Lefschetz numbers of iterations into sequences which can be locally realized as fixed point indices
of iterations at an isolated periodic orbit for some C 1 map. As a result,
to find the exact value of Drm [f ] we need two types of data: the information about the form of {L(f n )}n|r (more precisely of the set of so-called
2000 Mathematics Subject Classification. Primary 37C25, 55M20; Secondary
37C05.
Key words and phrases. Least number of periodic points, indices of iterations,
smooth maps, low dimensional dynamics.
Research supported by KBN grant No 1 P03A 03929.
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GRZEGORZ GRAFF AND JERZY JEZIERSKI
P
algebraic periods: {n ∈ N : k|n µ(n/k)L(f k ) 6= 0}, for n|r, where µ is
the Möbius function) and the description of all possible sequences of local
indices of iterations. The article joins the ideas of three papers: [5] in
which Drm [f ] is defined, [12] where the description of algebraic periods of
self-maps of S 3 is given, and finally [6], where there is a classification of
sequences of local indices of iterations in dimension 3. Basing on these
results we are able to determine Dr3 [f ] for all self-maps of a smooth 3manifold M which is closed connected and simply-connected, i.e., M is
3-dimensional sphere, provided that the result of G. Perelman on Poincaré
conjecture is true (cf. for example [8]).
It is worth pointing out that Dr3 [f ] is almost independent on f , i.e., is
insensitive on a homotopy class of f , which seems to be rather unexpected
conclusion. For example if r is odd and f is a self-map of S 3 of degree
with the modulus greater than 1, then Dr3 [f ] ∈ {ζ(r) − 1, ζ(r)}, where
ζ(r) is the number of all divisors of r (cf. Theorem 4.2). This is a result
of simply-connectedness of S 3 and of the fact that the set of algebraic
periods is equal to the set of all natural numbers. As a consequence, for 3dimensional sphere S 3 the value of Dr3 [f ] may be perceive as the invariant
of the whole space rather than the class of homotopy determined by f .
At present, our method of the calculation of the invariant Dr3 [f ] is limited to 3-dimensional manifolds, because there is no description of fixed
point indices of iterations at an isolated fixed point in higher dimensions.
In [5] the authors determined Dr3 [f ] for self-maps of S 2 × I. The further
development in this direction, i.e., finding the invariant for 3-manifolds
with boundary depends on the possibility of obtaining the overall description of algebraic periods of self-maps of these manifolds.
The article is organized as follows: in Sections 2 and 3 we introduce
the notation and definitions as well as the statements of [5], [6] and [12]
which will be used in the further part of the paper. In Section 4 we give
the main results (Theorem 4.2 and Theorem 4.6). The case of odd r is a
straightforward consequence of the results established in [5], [6], [12] (in
particular Theorem 2.9 from [5]), while the case of even r needs careful
and detailed analysis.
2. Preliminary results
A sequence of indices of iterations at an isolated fixed (periodic) point
plays crucial role in minimalization of the number of periodic points in
homotopy class. Let f : U → Rm , where U is an open subset of Rm , be a
map such that x0 is an isolated fixed point for each iteration of f . Then
the sequence of local indices {ind(f n , x0 )}∞
n=1 is well-defined. Below we
introduce useful notation of representing such sequences, which will be
used in the next sections.
MINIMAL NUMBER OF PERIODIC POINTS FOR SELF-MAPS OF S 3
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Definition 2.1. For a given k we define the basic sequence:
½
k if
k|n,
regk (n) =
0 if k 6 |n.
Each regk is an elementary periodic sequence of the form:
(0, . . . , 0, k, 0, . . . , 0, k, . . .),
where the non-zero entries appear for indices divisible by k.
A sequence of indices of iterations (as well as any integer sequence) may
be written in the form of so-called periodic expansion (cf. [13]), namely:
(2.1)
ind(f n , x0 ) =
∞
X
ak regk (n),
k=1
P
where an = n1 k|n µ(k)ind(f (n/k) , x0 ), µ is the classical Möbius function, i.e., µ : N → Z is defined by the following three properties: µ(1) = 1,
µ(k) = (−1)s if k is a product of s different primes, µ(k) = 0 otherwise.
It appears that indices of iterations must satisfy some conditions, found
in [4], called Dold relations (or Dold congruences).
Theorem 2.2. (Dold relations) All coefficients of the periodic expansion
for a sequence of indices of iterations are integers, i.e., in the formula
(2.1) ak ∈ Z.
S
For p ≥ 1 we define Pp (f ) = Fix(f p ) \ 0<n<p Fix(f n ). If x ∈ Pp (f ),
then the orbit of x will be called p-orbit.
Now we introduce the notion of a Differential Dold sequence in Rm for
a p-orbit, which we will denote as DDm (p) sequence. This is a sequence
which can be realized as a sequence of indices of iterations on isolated
p-orbit for some smooth map in m-dimensional space.
m
Definition 2.3. A sequence of integers {cn }∞
n=1 is called a DD (p) sequence if there is an isolated p-orbit P , its neighborhood U ⊂ Rm and a
C 1 map φ : U → Rm such that cn = ind(φn , P ). If this equality holds
for n|r, where r is fixed, then the finite sequence {cn }n|r will be called
DDm (p|r) sequence. ¤
There is a strict relation, established in [5], between the minimal number of r-periodic points for all smooth maps in a given homotopy class
and DDm (p|r) sequences (Theorem 2.5 below).
Definition 2.4. Let {ξn }n|r be a sequence of integers satisfying Dold relations, i.e., its coefficients in the periodic expansion are integers. Assume
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GRZEGORZ GRAFF AND JERZY JEZIERSKI
that we are able to decompose {ξn }n|r into the sum:
ξ(n) = c1 (n) + ... + cs (n),
m
where ci is a DD (li |r) sequence for i = 1, ..., s. Each such decomposition
determine the number l = l1 + ... + ls . We define the number Drm [ξ] as
the smallest l which can be obtained in this way.
¤
Let {L(f n )}n|r be the sequence of Lefchetz number of iterations, we
will denote Drm [f ] = Drm [{L(f n )}n|r ].
Theorem 2.5. ([5]) Let M be a smooth compact connected and simplyconnected manifold of dimension m ≥ 3 and r ∈ N a fixed number. For
M with nonempty boundary we assume additionally that f has no periodic
points on the boundary. Then,
min{#Fix(g r ) : g is C1 and is homotopic to f } = Drm [f ].
¤
By Definition 2.4 determining Drm [f ] demands the knowledge of the
forms of all DDm (p) sequences. There are strong restrictions on such
sequences, found by Chow, Mallet-Paret and Yorke (cf. [3]).
It is not difficult to observe that in order to get any DDm (p) sequence
it is enough to replace each ak regk by ak regpk in the periodic expansion
of some DDm (1) sequence (we will say that this DDm (p) sequence comes
from the given DDm (1) sequence). As a consequence, we will be able to
calculate Drm [f ] if we know all the forms of DDm (1) sequences. This is
provided in dimension 3 by the following theorem:
Theorem 2.6. ([6])
There are seven kinds of DD3 (1) sequences:
(A) cA (n) = a1 reg1 (n) + a2 reg2 (n),
(B) cB (n) = reg1 (n) + ad regd (n),
(C) cC (n) = −reg1 (n) + ad regd (n),
(D) cD (n) = ad regd (n),
(E) cE (n) = reg1 (n) − reg2 (n) + ad regd (n),
(F) cF (n) = reg1 (n) + ad regd (n) + a2d reg2d (n), where d is odd,
(G) cG (n) = reg1 (n) − reg2 (n) + ad regd (n) + a2d reg2d (n), where d is
odd.
In all cases d ≥ 3 and ai ∈ Z.
In fact in dimension 3 we will need to know only the forms of some
special DD3 (2) sequences, in addition to DD3 (1) sequences, for finding
the value of Drm [f ] (see Lemma 2.7).
Let us list below three DD3 (2) sequences which come from DD3 (1)
sequences of the form (E), (F) and (G):
MINIMAL NUMBER OF PERIODIC POINTS FOR SELF-MAPS OF S 3
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(E’) cE 0 (n) = reg2 (n) − reg4 (n) + a2d reg2d (n), where d ≥ 3,
(F’) cF 0 (n) = reg2 (n) + a2d reg2d (n) + a4d reg4d (n), where d ≥ 3 is odd,
(G’) cG0 (n) = reg2 (n) − reg4 (n) + a2d reg2d (n) + a4d reg4d (n), where
d ≥ 3 is odd.
Lemma 2.7. ([5]) Let f : M → M be a C 1 map, dim M = 3, then in the
definition of Dr3 [f ] it is enough to consider only DD3 (1|r) sequences, i.e.,
sequences which for n|r are of the forms (A)-(G) and DD3 (2|r) sequences
of the forms (E’), (F’) and (G’).
Let r be a fixed natural number. The sequence of Lefschetz numbers
{L(f n )}∞
n=1 also satisfies Dold relations, so we can write it down in the
form of a periodic expansion:
(2.2)
n
L(f ) =
∞
X
bk regk (n).
k=1
Definition 2.8. For a given periodic expansion (2.2) of {L(f n )}∞
n=1 , we
define B(f ), the set of algebraic periods of f , as B(f ) = {k ∈ N : bk 6= 0}
and Br (f ), the set of algebraic periods of f up to the level r, as Br (f ) =
{k ∈ N : k|r and bk 6= 0}.
Let us now rewrite the formula (2.2) for n|r into:
(2.3)
L(f n ) = b1 reg1 (n) + b2 reg2 (n) + b4 reg4 (n) +
X
bk regk (n),
k∈G
where b1 , b2 , b4 are arbitrary integers,
G = {k ∈ N : k 6∈ {1, 2, 4} and bk 6= 0} = Br (f ) \ {1, 2, 4}.
Let us define H, the subset of G:
H = {k ∈ G : k is odd and bk 6= 0, b2k 6= 0}.
In the next section we will use the following result proved in [5]:
Theorem 2.9. If r is odd, then:
½
#G
3
(∗) Dr [f ] =
#G + 1
if |L(f )| ≤ #G,
otherwise.
If r is even and r > 4, then:
(∗∗) Dr3 [f ] ∈ [#G − #H, #G − #H + 2].
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GRZEGORZ GRAFF AND JERZY JEZIERSKI
3. Algebraic periods
In order to calculate Dr3 [f ] we need to know the decomposition of
Lefschetz numbers of iterations into the periodic expansion. Thus, we
need the exact form of the set of algebraic periods of f (cf. Definition
2.8). We will base on the description of algebraic periods for self-maps of
S 3 which is given in [12].
We use homology spaces with the coefficients in the field of rational
numbers Q.
Let us recall that
½
3
Hi (S ) =
Q for i = 0, 3;
0 otherwise.
Let f : S 3 → S 3 . The homomorphism f∗3 : H3 (S 3 ; Q) → H3 (S 3 ; Q) is
a multiplication by a number β ∈ Z, called a degree of f (denoted also as
deg(f )). Assume r ∈ N is fixed. We consider the periodic expansion of
{L(f n )}∞
n=1 given by (2.2). Let us remind to the reader that by Definition
2.8 bn 6= 0 is equivalent to the statement that n ∈ B(f ), i.e., that n is an
algebraic period.
Lemma 3.1. ([12], Theorem 1.2) Let bn denote the nth coefficient of the
periodic expansion of {L(f n )}∞
n=1 then:
(a) b1 = 1 − β.
(b) b2 = 0 if and only if β ∈ {0, 1}.
(c) If n > 2, then bn = 0 if and only if β ∈ {−1, 0, 1}.
Let ζ(r) denote the number of all divisors of r. Assume that β 6∈
{−1, 0, 1}. Then, by Lemma 3.1 B(f ) = N and the following Proposition
holds:
Proposition 3.2. Br (f ) = {n ∈ N : n|r}, or equivalently #Br (f ) =
ζ(r).
By Proposition 3.2 we obtain:
Proposition 3.3.
(1) For r odd G = Br (f ) \ {1}, thus #G = ζ(r) − 1.
(2) For r even there is:
if 4|r, then G = Br (f ) \ {1, 2, 4}, thus #G = ζ(r) − 3,
if 4 6 |r, then G = Br (f ) \ {1, 2}, thus #G = ζ(r) − 2.
Lemma 3.4. If r is even, then #H= η(r) − 1, where η(r) denotes the
number of all odd divisors of r.
MINIMAL NUMBER OF PERIODIC POINTS FOR SELF-MAPS OF S 3
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Proof. Observe that #H is the number of pairs k, 2k, where k|r, k > 1
is odd, in G. As each natural k is an algebraic period, every odd k > 1
determines such a pair and thus an element in H.
¤
4. Minimal number of periodic points for self-maps of S 3
The complete description of the minimal number of r-periodic points
for all smooth maps homotopic to the map f : S 3 → S 3 of degree β will
be given in Proposition 4.1 and Theorems 4.2 and 4.6 below.
Proposition 4.1. For each self-map f of S 3 we have: L(f n ) = 1 − β n .
This implies the following statements.
If β = 1, then L(f n ) = 0 for all n, hence Dr3 [f ] = 0.
If β = −1, then L(f n ) = 2reg1 (n) − reg2 (n), hence Dr3 [f ] = 1 for all
r, because we can realize Lefschetz numbers of iterations by one sequence
of the type (A).
If β = 0, L(f n ) = reg1 (n), and analogously as in the previous case,
3
Dr [f ] = 1.
¤
In order to find Dr3 [f ], for f such that |β| = |deg(f )| > 1, we will use
Theorem 2.9. In the case of odd r, Theorem 2.9 with Proposition 3.3 part
(1) and the fact that L(f ) = 1 − β give immediately the exact value of
Dr3 [f ]:
Theorem 4.2. For r odd, |β| > 1 there is:
½
ζ(r) − 1 if ζ(r) ≥ β ≥ 2 − ζ(r),
Dr3 [f ] =
ζ(r)
otherwise.
4.1. The case of even r. Let us consider the minimal realization of
Lefschetz numbers of iterations which consists
P of a set A of sequences {ci },
where each ci is DD3 (pi |r) sequence and i pi = Dr3 [f ] (cf. Definition
2.4).
(4.1)
X
X
L(f n ) = b1 reg1 (n) + b2 reg2 (n) + b4 reg4 (n) +
bk regk (n) =
ci (n).
k∈G
i
Note that, by Lemma 2.7, each ci has one of the forms (A)-(G), (E’)(G’) which implies that pi ≤ 2. Let us remind to the reader that by
Theorem 2.9:
(4.2)
Dr3 [f ] ∈ [#G − #H, #G − #H + 2].
For f , a self-map S 3 , the following lemma holds.
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GRZEGORZ GRAFF AND JERZY JEZIERSKI
Lemma 4.3. If |β| = |deg(f )| > 1, 4|r and r > 4 then each set of
sequences A realizing Dr3 [f ] contains a sequence of the type (B)-(E) with
the basic sequence a4 reg4 .
Proof. Suppose that in A there is no sequence of one of the types (B)2
4
≤ −3, for |β| > 1, which implies the
(E) with a4 reg4 . Then b4 = β −β
4
existence of three sequences:
(4.3)
γ1 , γ2 , γ3
of the type (G’) or (E’), since only these give the negative contribution to
b4 . We will show that this leads to contradiction with minimality of A.
For odd d (d > 1, 4d|r) let us consider the following triple which appears
in the formula (4.1):
(4.4)
bd regd (n) + b2d reg2d (n) + b4d reg4d (n).
By Lemma 3.1 each coefficient in (4.4) is non-zero. A sequence (G’) may
be used only if it reduces a part of such triple, namely its two last terms.
Then we have to use one sequence (B)-(E) to reduce bd regd , which gives
the contribution of the triple (4.4) to Dr3 [f ] equal to 2 + 1 = 3. On the
other hand, we may use one (F) or (G) to reduce two first terms and
one sequence (B)-(E) to reduce the last. As a result we get the smaller
contribution 1 + 1 = 2.
Suppose that among γ1 , γ2 , γ3 in (4.3) there are three (G’)’s. They give
the contribution to Dr3 [f ] equal to 9. On the other hand, independently on
the values of b1 , b2 , b4 , there is a smaller realization: we may reduce these
3 triples by 6 other sequences in the way indicated above and b1 reg1 (n) +
b2 reg2 (n) + b4 reg4 (n) by one sequence (A) and one (B)-(E). This gives 8
sequences in total, contradicting to minimality.
Similarly, we get a contradiction when we assume that at least one of
the three sequences γ1 , γ2 , γ3 is of the type (E’) (we replace the expression
cE 0 = reg2 (n) − reg4 (n) + a2d reg2d (n) which counts with the multiplicity
2 with cD = a2d reg2d (n) of multiplicity 1 and repeat the same reasoning
as in the case of 3 sequences (G)).
¤
Corollary 4.4. Under the assumptions of Lemma 4.3
Dr3 [f ] ∈ [#G − #H + 1, #G − #H + 2].
Proof. By Theorem 2.9 it is enough to exclude the possibility Dr3 [f ] =
#G − #H. Assume that A realizes Dr3 [f ] = #G − #H. By Lemma
4.3 there is a sequence of one of the types (B)-(E) in A with the term
a4 reg4 . The remaining #G − #H − 1 sequences (counting multiplicity)
must reduce bk regk for k ∈ G. Only sequences (F) and (G) reduce two
bk regk , k ∈ G and we may use them in such a way #H times ((F’), (G’)
MINIMAL NUMBER OF PERIODIC POINTS FOR SELF-MAPS OF S 3
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also reduce two bk regk ’s but they are counted twice). Now the remaining
#G − 2#H bk regk ’s must be reduced by #G − 2#H − 1 sequences (A)(E), (E’)-(G’) (counting multiplicity). This is impossible, since each of
(A)-(E) reduces at most one bk regk with k ∈ G, and (E’)-(G’) is counted
twice and reduces at most two bk regk with k ∈ G.
¤
Lemma 4.5. Suppose that |deg(f )| > 1 and Dr3 [f ] = #G−#H +1. Then
there is a minimal set realizing Dr3 [f ] containing only sequences (B)-(G).
Moreover we may assume that this set contains one sequence of type
(B)-(E) with ad regd such that d = 4 and #G − 2#H sequences of the type
(B)-(E) with d 6= 4 and #H sequences of the type (F)-(G).
¤
Proof. Let us fix a minimal set A. By Lemma 4.3 there is a sequence c(4)
in A of one of the types (B)-(E) with a4 reg4 . We take then a4 = b4 . The
remaining sequences realize bk regk for k ∈ G, hence each of them must
realize at least one bk regk so none of them is of the type (A).
Let us assume that A contains a sequence of the type (G’)
cG0 = reg2 − reg4 + a2d reg2d + a4d reg4d .
We will show that it is possible to exchange A into another minimal
system A0 in which cG0 does not appear. Namely, we change 3 sequences
in A:
• instead of cG0 we take two sequences of the type (D): a2d reg2d ,
a4d reg4d ;
• instead of c(4) with a4 reg4 we take c0(4) with (a4 − 1)reg4 .
2
• Let us notice that b2 = β−β
< 0 implies the existence of a
2
sequence ψ(n) of the type (E) or (G), since only these have
negative contribution to b2 . Instead of such sequences we take
ψ 0 (n) = reg2 + ψ(n), which is an expression of the type (B) or
(F) respectively.
This gives a system A0 with the same multiplicity as A (so also minimal), with one expression of the type (G’) less. Similarly we may remove
expressions of type (E’) and (F’).
¤
Theorem 4.6. Let r be even, |β| > 1, then Dr3 [f ] ∈ {ζ(r)−η(r)−1, ζ(r)−
η(r)}. What is more, Dr3 [f ] = ζ(r) − η(r) − 1 if and only if one of the
two following conjunctions holds:
(4.5)
(i) ζ(r) − η(r) ≥ β 2
(ii) ζ(r) − 3η(r) ≥ β − 2
and
and
β 2 −β
2
β 2 −β
2
− η(r) ≥ 0,
− η(r) ≤ −1.
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GRZEGORZ GRAFF AND JERZY JEZIERSKI
Before we give the proof of the above theorem, we illustrate it by the
following example:
Example 4.7. Let f : S 3 → S 3 have the degree β = 2 and let r = 12.
The Lefschetz numbers L(f n ) are equal to 1 − 2n . We represent this
sequence (for n|12) in the form of the periodic expansion. We get that
2
2
4
b1 = 1 − β = −1, b2 = β−β
= −1, b4 = β −β
= −3, thus:
2
4
(4.6)
L(f n ) = −reg1 (n)−reg2 (n)−3reg4 (n)+b3 reg3 (n)+b6 reg6 (n)+b12 reg12 (n).
There is: ζ(12) = 6, η(12) = 2. Because ζ(r) − 3η(r) = β − 2 = 0 and
β 2 −β
− η(r) = −1, we see that the second condition of (4.5) is satisfied
2
and thus Dr3 [f ] = ζ(r) − η(r) − 1 = 3. Indeed, we may take the 3 following
sequence, which in sum realize {L(f n )}n|12 :
cG (n) = reg1 (n) − reg2 (n) + b3 reg3 (n) + b6 reg6 (n),
cC1 (n) = −reg1 (n) − 3reg4 (n),
cC2 (n) = −reg1 (n) + b12 reg12 (n).
Proof of Theorem 4.6.
First notice that if |β| > 1, then D23 (f ) = 1 and D43 (f ) = 2, thus we
verify straightforwardly that for r = 2 and r = 4 the theorem holds. For
r > 4 we divide the proof into two cases.
PART (I): 4|r.
By Corollary 4.4 Dr3 [f ] ∈ {#G−#H +1, #G−#H +2}, which is equal,
by item (2) of Proposition 3.3 for 4|r, to {ζ(r) − η(r) − 1, ζ(r) − η(r)}.
We will now find the conditions equivalent to the statement that Dr3 [f ] =
#G − #H + 1 = ζ(r) − η(r) − 1.
By Lemma 4.5 we may assume that there is a minimal set of sequences
A realizing Dr3 [f ] with #H sequences (F) or (G) and #G−2#H sequences
(B)-(F), which reduce bk regk for k 6= 4, and one extra expression c(4) of
the type (B), (C), (D) or (E), which reduces b4 reg4 . Thus there are
#G − #H sequences plus one extra in A.
Let us denote the contribution of the single sequence c(4) to the first
two terms of the formula (4.1) by ²1 reg1 (n) + ²2 reg2 (n), where (²1 , ²2 ) ∈
{(1, 0), (−1, 0), (0, 0)(1, −1)}.
By mX , where X ∈ {B, C, D, E, F, G} we denote the number of sequences (not counting c(4) ) of the given type in the minimal realization A. Then Dr3 [f ] = #G − #H + 1 if and only if there are integers
MINIMAL NUMBER OF PERIODIC POINTS FOR SELF-MAPS OF S 3
11
(²1 , ²2 ) ∈ {(1, 0), (−1, 0), (0, 0)(1, −1)} such that the following system of
equations has an integer solution in non-negative unknowns mX :
(4.7)
mB
mB
+mC
−mC
+mD
+mE
+mE
−mE
+mF
+mF
+mF
+mG
+mG
−mG
+mG
= #G − #H,
= b1 − ²1 ,
= b2 − ²2 ,
= #H,
where the first and last equations describe the number of sequences
(not counting c(4) ), the second and third - their contribution to the first
two terms of periodic expansion.
The above system is equivalent to:
mD
(4.8)
+mB
+mB
+mF
+mF
+mF
+mG
+mG
+mG
+mG
= #G − #H − mC − mE ,
= b1 + mC − mE − ²1 ,
= #H,
= −b2 − mE + ²2 .
Notice that:
• for any fixed mC , mE (4.8) is a Cramer system with the determinant +1, thus mD , mB , mF , mG are uniquely determined.
• If mC , mE are integers, then the other unknowns must be integers.
• For any fixed values of mC ≥ 0, mE ≥ 0 the solutions of (4.8)
are non-negative if and only if the following system of inequalities
holds:
(4.9)
0 ≤ −b2 − mE + ²2 ≤ #H ≤ b1 + mC − mE − ²1 ≤ #G − #H − mC − mE .
As a consequence of the above facts, in order to find the solution of the
system (4.7), it is enough to solve (4.9) with the unknowns mC , mE , such
that mC ≥ 0, mE ≥ 0. We rewrite (4.9) in the system of four inequalities:
(4.10)
mE ≤ −b2 + ²2 ,
(4.11)
mE ≥ −b2 + ²2 − #H,
(4.12)
mC − mE ≥ #H − b1 + ²1 ,
1
(−b1 + #G − #H + ²1 ).
2
The problem of finding the needed solution of (4.7) reduces to the
question if there is a point in the plane (mC , mE ) ∈ Z2 , mC ≥ 0, mE ≥ 0
for which inequalities (4.10)-(4.13) are satisfied.
(4.13)
mC ≤
12
GRZEGORZ GRAFF AND JERZY JEZIERSKI
We substitute the values of #G and #H using Lemma 3.3 part (2) and
Lemma 3.4 and the values of b1 and b2 calculated straightforwardly:
1
#G = ζ(r) − 3; #H = η(r) − 1; b1 = 1 − β; b2 = (β − β 2 ).
2
Then the inequalities (4.10)-(4.13) can be rewritten as:
(4.14)
(4.15)
(4.16)
(4.17)
1
(ζ(r) − η(r) + β − 3 + ²1 ),
2
mC ≤
mE ≥
1 2
(β − β) − η(r) + 1 + ²2 ,
2
mE ≤
1 2
(β − β) + ²2 ,
2
mE ≤ mC − η(r) − β + 2 − ²1 .
Now the problem becomes: for which numbers: r ∈ N, β ∈ Z (4|r ,
|β| ≥ 2) one can choose parameters (²1 , ²2 ) ∈ {(0, 0), (1, 0), (−1, 0), (1, −1)}
so that the inequalities (4.14) - (4.17) have a non-negative integer solution
(mC , mE ). To simplify the notations we denote:
1
b = (ζ(r) − η(r) + β − 3 + ²1 ),
2
1
c = (β 2 − β) − η(r) + 1 + ²2 ,
2
1
d = (β 2 − β) + ²2 ,
2
e = −η(r) − β + 2 − ²1 .
Now the system of inequalities (4.14) - (4.17) takes the form:
(4.18)
mC ≤ b; c ≤ mE ; mE ≤ d; mE ≤ mC + e.
Lemma 4.8. Let b, c, d, e ∈ R satisfy: c, e ∈ Z and c ≤ d. Then the
inequalities (4.18) have a non-negative integer solution (mC , mE ) ⇐⇒
b ≥ 0, d ≥ 0, max{c, 0} ≤ [b] + e, where [b] denotes the integer part of
b.
In order to proof the above lemma it is enough to notice that the first
three inequalities give (−∞, b > × < c, d > while the last defines the
half-plane under the line mE = mC + e in (mC , mE ) coordinates.
Finally by Lemma 4.8 the problem becomes: for which β and r one
can choose such parameters (²1 , ²2 ) that the following inequalities hold.
(4.19)
ζ(r) − η(r) + β − 3 + ²1 ≥ 0,
MINIMAL NUMBER OF PERIODIC POINTS FOR SELF-MAPS OF S 3
(4.20)
13
1 2
(β − β) + ²2 ≥ 0,
2
(4.21)
1
1
max{ (β 2 −β)−η(r)+1+²2 , 0} ≤ [ (ζ(r)−η(r)+β−3+²1 )]−η(r)−β+2−²1 .
2
2
We notice that the inequality (4.20) always hold. In fact |β| ≥ 2 implies
β 2 − β ≥ 2 and 21 (β 2 − β) + ²2 ≥ 1 − 1 ≥ 0.
Now we study (4.19). We will consider four cases.
Case (1) ζ(r) − η(r) + β ≤ 1. Then the inequality (4.19) never holds,
hence the system has no solution.
Case (2) ζ(r) − η(r) + β = 2 and Case (3) ζ(r) − η(r) + β = 3
will be discussed individually.
Case (4) ζ(r) − η(r) + β ≥ 4. Then the inequality (4.19) holds for
each ²1 .
We will consider these cases (in reverse order: starting from Case 4 to
Case 1) as the assumptions in the next subcases. We will look for the
solutions of the inequality (4.21).
Case (4) We assume that ζ(r)−η(r)+β ≥ 4 (the inequality (4.19) holds
for each ²1 ). To release from maximum and the integer part in (4.21) we
consider several subcases.
Subcase (4.≥) 21 (β 2 − β) − η(r) ≥ 0.
Now c = 12 (β 2 − β) − η(r) + 1 − ²2 ≥ 0, since ²2 ≥ −1. The inequality
(4.21) takes the form:
1 2
1
(β − β) − η(r) + 1 + ²2 ≤ [ (ζ(r) − η(r) + β − 3 + ²1 )] − η(r) − β + 2 − ²1 ,
2
2
or
1 2
1
(β + β) − 1 + ²1 + ²2 ≤ [ (ζ(r) − η(r) + β − 3 + ²1 )].
2
2
We notice that if the above inequality holds for a (²1 , ²2 ) ∈ {(0, 0), (1, 0), (−1, 0), (1, −1)}
then it also holds for (−1, 0), thus it is enough to solve
1 2
1
(β + β) − 2 ≤ [ (ζ(r) − η(r) + β − 4)],
2
2
hence
1 2
1
(β + β) ≤ [ (ζ(r) − η(r) + β)].
2
2
Subsubcase (4.≥.even) ζ(r) − η(r) + β is even.
Now we may leave the integer part:
1
1 2
(β + β) ≤ (ζ(r) − η(r) + β),
2
2
which implies
β 2 ≤ ζ(r) − η(r).
14
GRZEGORZ GRAFF AND JERZY JEZIERSKI
Subsubcase (4.≥.odd) ζ(r) − η(r) + β is odd.
Now we get:
1
1
1 2
(β + β) ≤ (ζ(r) − η(r) + β) − ,
2
2
2
which implies
β 2 ≤ ζ(r) − η(r) − 1.
Moreover we notice that in this subsubcase the above inequality is equivalent to
β 2 ≤ ζ(r) − η(r).
In fact, by the parity assumption in this subsubcase we have that ζ(r) −
η(r) + β ≡ 1 (mod 2), and thus ζ(r) − η(r) − β 2 ≡ 1 (mod 2). As a
consequence the equality β 2 = ζ(r) − η(r) can not hold.
Thus the assumptions of Case 4, Subcase (4.≥) and the obtained inequality give the following system of conditions:
(4.22)
2
ζ(r) − η(r) + β ≥ 4 and β 2−β − η(r) ≥ 0 and
ζ(r) − η(r) ≥ β 2 .
Subcase (4.<) 21 (β 2 − β) − η(r) < 0.
Now c = 12 (β 2 − β) − η(r) + 1 + ²2 ≤ 0 (for any ²2 ) hence inequality
(4.21) becomes
1
0 ≤ [ (ζ(r) − η(r) + β − 3 + ²1 )] − η(r) − β + 2 − ²1 .
2
Let us notice that if the inequality holds for an −1 ≤ ²1 ≤ +1, then it
also holds for ²1 = −1, hence we get
1
η(r) + β − 3 ≤ [ (ζ(r) − η(r) + β) − 2].
2
Subsubcase (4.<.even) ζ(r) − η(r) + β is even.
Now η(r) + β − 3 ≤ 21 (ζ(r) − η(r) + β) − 2 or
ζ(r) − 3η(r) ≥ β − 2.
Subsubcase (4.<.odd) ζ(r) − η(r) + β is odd.
Now η(r) + β − 3 ≤ 21 (ζ(r) − η(r) + β) − 2 − 21 hence we get
ζ(r) − 3η(r) ≥ β − 1.
Moreover we notice that in this subsubcase the above inequality is equivalent to
ζ(r) − 3η(r) ≥ β − 2.
In fact the equality ζ(r)−3η(r) = β −1 can not hold because of the parity
assumptions in (4.<.odd).
MINIMAL NUMBER OF PERIODIC POINTS FOR SELF-MAPS OF S 3
15
Thus, Case 4, Subcase (4.<) and the obtained inequality give the following system of conditions:
(4.23)
2
ζ(r) − η(r) + β ≥ 4 and β 2−β − η(r) ≤ −1 and ζ(r) − 3η(r) ≥ β − 2.
Case (3) ζ(r) − η(r) + β = 3.
The assumption of Case 3 implies that
1
1
b = (ζ(r) − η(r) + β − 3 + ²1 ) = ²1 .
2
2
Now b ≥ 0 for ²1 = 0 or +1.
Subcase (3.0) ²1 = 0.
Since ²1 = 0 implies ²2 = 0, the inequality (4.21) takes the form
1
max{ (β 2 − β) − η(r) + 1, 0} ≤ −η(r) − β + 2.
2
Subsubcase (3.0.≥) 12 (β 2 − β) − η(r) + 1 ≥ 0.
Now we get
1 2
(β − β) − η(r) + 1 ≤ −η(r) − β + 2,
2
which is equivalent to
β2 + β ≤ 2
and the last holds only for β = −2. Then the conditions (3.0.≥) and of
Case (3) take the form: 4 ≥ η(r) and ζ(r) = η(r) + 5 respectively. Since
η(r)|ζ(r), η(r) = 1 and ζ(r) = 6. This implies r = 25 .
Subsubcase (3.0.<) 12 (β 2 − β) − η(r) + 1 < 0. We get:
0 ≤ −η(r) − β + 2.
In other words η(r) ≤ −β + 2. On the other hand the assumption (3.0.<)
gives β 2 − β < 2η(r) − 2. The above inequalities imply β 2 + β − 2 < 0,
which is never true for |β| > 1.
Subcase (3.1) ²1 = 1.
1
max{ (β 2 − β) − η(r) + 1 + ²2 , 0} ≤ −η(r) − β + 1.
2
Since ²2 may be 0 or −1, here we may put ²2 = −1 which implies
1
max{ (β 2 − β) − η(r), 0} ≤ −η(r) − β + 1.
2
Subsubcase (3.1.≥) 12 (β 2 − β) − η(r) ≥ 0.
Now
1 2
(β − β) − η(r) ≤ −η(r) − β + 1
2
16
GRZEGORZ GRAFF AND JERZY JEZIERSKI
implies β 2 + β − 2 ≤ 0, hence β = −2. Then the assumption of Case 3:
ζ(r) − η(r) + β = 3 gives ζ(r) = η(r) + 5. On the other hand the condition
(3.1.≥) takes the form: 3 ≥ η(r). Again η(r)|ζ(r) implies η(r) = 1,
ζ(r) = 6 and thus r = 25 .
Subsubcase (3.1.<) 12 (β 2 − β) − η(r) < 0.
The inequality (4.21) takes the form
0 ≤ −η(r) − β + 1,
hence implies η(r) ≤ −β + 1. We join this inequality with the condition
(3.1.<) and get
1 2
(β − β) < η(r) ≤ −β + 1.
2
This implies the inequality
1 2
(β − β) < −β + 1,
2
which is not valid for any |β| > 12.
Case (2)
ζ(r) − η(r) + β = 2.
Here we get:
1
1
(ζ(r) − η(r) + β − 3 + ²1 ) = (²1 − 1).
2
2
Notice that b ≥ 0 only for ²1 = +1 and then b = 0. This yields (4.21) has
the form
1
max{ (β 2 − β) − η(r) + 1 + ²2 , 0} ≤ −η(r) − β + 1.
2
Since ²2 may be 0 or −1, it is enough to consider ²2 = −1. We get then:
b=
1
max{ (β 2 − β) − η(r), 0} ≤ −η(r) − β + 1.
2
Subcase (2.≥) 21 (β 2 − β) − η(r) ≥ 0.
The inequality (4.21) takes the form
1 2
(β − β) − η(r) ≤ −η(r) − β + 1,
2
which implies β 2 + β ≤ 2 hence β = −2. Moreover the conditions (2.≥)
and of Case (2) become: 3 ≥ η(r) and ζ(r) = η(r) + 4 respectively. Since
η(r)|ζ(r), there are two possibilities: η(r) = 1 or η(r) = 2. In the first
case ζ(r) = 5 and r = 24 . In the second case ζ(r) = 6 and r = 22 p where
p is a prime number greater than 2. Now we get either
• β = −2 and r = 24 or
• β = −2 and r = 4p for an odd prime number p.
MINIMAL NUMBER OF PERIODIC POINTS FOR SELF-MAPS OF S 3
17
Subcase (2.<) 21 (β 2 − β) − η(r) < 0.
Now the inequality (4.21) takes the form 0 ≤ −η(r) − β + 1, which
implies η(r) ≤ −β + 1. On the other hand the condition (2.<) gives
β 2 − β < 2η(r). This implies β 2 + β < 2 which has no solution for |β| > 1.
Summing up the previous considerations, we get that Dr3 [f ] = ζ(r) −
η(r) − 1 if and only if one of the five following conjunctions hold (cf.
(4.22), (4.23), Cases (3) and (2)):
(4.24)
β 2 −β
(1) ζ(r) − η(r) + β ≥ 4 and ζ(r) − η(r) ≥ β 2
and
− η(r) ≥ 0,
2
β 2 −β
(2) ζ(r) − η(r) + β ≥ 4 and ζ(r) − 3η(r) ≥ β − 2 and
− η(r) ≤ −1,
2
(3) β = −2
and r = 16,
(4) β = −2
and r = 32,
(5) β = −2
and r = 4p for an odd
prime p.
We show now that the system of conditions (4.24) is equivalent to the
conditions (i) and (ii) in Theorem 4.5, which we remind below:
(4.25)
(i) ζ(r) − η(r) ≥ β 2
(ii) ζ(r) − 3η(r) ≥ β − 2
and
and
β 2 −β
2
β 2 −β
2
− η(r) ≥ 0,
− η(r) ≤ −1.
We have to show that
[(i) or (ii)] ⇐⇒ [(1) or (2) or (3) or (4) or (5)].
First we prove that (ii) ⇐⇒ (2).
⇐ is trivial. To prove ⇒ it remains to show that the second at the third
inequalities in (2) imply the first. The second and third inequality give
2
respectively ζ(r) − η(r) ≥ 2η(r) + β − 2 and η(r) ≥ β 2−β + 1. This implies
that ζ(r) − η(r) ≥ β 2 , and as β 2 ≥ 4 − β for β 6= −2, we get that the first
inequality results from the second and the third for β 6= −2. If β = −2, we
get η(r) ≥ 4 and we should check whether ζ(r) − η(r) ≥ 2η(r) − 4 implies
ζ(r) − η(r) ≥ 6. This is obviously satisfied if η(r) > 4, and for η(r) = 4
(because 4|r) we get that ζ(r) must be greater than 10 so ζ(r) − η(r) ≥ 6
is in this case also satisfied.
We obtained that (2) ⇐⇒ (ii).
Now, to prove the Theorem, it remains to show that
(i) ⇐⇒ [(1) or (3) or (4) or (5)].
⇐ (1) implies (i) in a trivial way. Then we check case by case that
each of (3), (4), (5) implies (i).
⇒ We show that (i) and the negation of (1) imply the alternative
[(3) or (4) or (5)].
18
GRZEGORZ GRAFF AND JERZY JEZIERSKI
Recall that (i) means
β2 − β
− η(r) ≥ 0.
2
Now the negation of (1) means that the first inequality in (1) is not true,
i.e.,
ζ(r) − η(r) + β < 4.
The two above inequalities imply: β 2 ≤ ζ(r) + η(r) < 4 − β hence β 2 +
β − 4 < 0 which holds only for β = −2. This in turn implies
ζ(r) − η(r) ≥ β 2 and
ζ(r) − η(r) ≥ 4 ; η(r) ≤ 3 ; ζ(r) − η(r) < 6.
Now we get (ζ(r) − η(r) = 4 or 5) and η(r) ≤ 3.
If ζ(r) − η(r) = 5 then η(r) = 1 hence r = 25 = 32, so we get the case
(4).
If ζ(r) − η(r) = 4 then η(r) = 1 or 2. For η(r) = 1 we get r = 24 = 16
so we obtain the case (3). Finally for η(r) = 2, r = 4p, where p is an odd
prime, which gives the case (5). This ends the proof of the equivalency of
(4.24) and (4.25) and the proof of Part I.
PART (II): 2|r but 4 6 |r.
Because b4 = 0, we see that Dr3 [f ] ∈ [#G − #H, #G − #H + 1] =
[ζ(r) − η(r) − 1, ζ(r) − η(r)], where the last equality results from Lemma
3.4 and item (2) of Proposition 3.3 for 4 6 |r.
Searching for the conditions equivalent to Dr3 [f ] = #G−#H, we repeat
the same reasoning as in the case 4|r, with the only difference which is due
to the fact that we do not need additional sequence to reduce b4 reg4 , so
we have to find non-negative integer solutions of (4.7) without parameters
²1 , ²2 .
Now we follow the previous part and taking into account that #G =
ζ(r) − 2, we get instead of the inequalities (4.19)-(4.21):
(4.26)
ζ(r) − η(r) + β − 2 ≥ 0,
(4.27)
1 2
(β − β) ≥ 0,
2
1
1
(4.28) max{ (β 2 −β)−η(r)+1, 0} ≤ [ (ζ(r)−η(r)+β−2)]−η(r)−β+2.
2
2
We notice that the inequality (4.27) is always satisfied for |β| > 1.
Since the inequality (4.26) is an explicit condition, it remains to solve
(4.28). Again we consider some subcases.
Case (>) 12 (β 2 − β) − η(r) + 1 > 0.
MINIMAL NUMBER OF PERIODIC POINTS FOR SELF-MAPS OF S 3
19
Subcase (>.even) ζ(r) − η(r) + β − 2 is even.
Then the inequality (4.28) becomes
1
1 2
(β − β) − η(r) + 1 ≤ (ζ(r) − η(r) + β − 2) − η(r) − β + 2,
2
2
which is equivalent to
β 2 ≤ ζ(r) − η(r).
Subcase (>.odd) ζ(r) − η(r) + β − 2 is odd. Then we get
β 2 ≤ ζ(r) − η(r) − 1.
Since by the parity assumption the equality β 2 = ζ(r)−η(r) can not hold,
the last inequality is equivalent to
β 2 ≤ ζ(r) − η(r).
(4.29)
Finally, as the inequality (4.26) follows from (4.29) we get the conjunction (i) of the conditions (4.5).
Case (≤) 12 (β 2 − β) − η(r) + 1 ≤ 0.
Subcase (≤.even) ζ(r) − η(r) + β − 2 is even.
The inequality (4.28) becomes
1
(ζ(r) − η(r) + β − 2) − η(r) − β + 2,
2
which is equivalent to
0≤
ζ(r) − 3η(r) ≥ β − 2.
Subcase (≤.odd) ζ(r) − η(r) + β is odd.
Then we get ζ(r) − 3η(r) ≥ β − 1. Again by the parity assumption we
may weaken the inequality putting
(4.30)
ζ(r) − 3η(r) ≥ β − 2.
As (4.30) and 12 β 2 − β) − η(r) + 1 ≤ 0 imply the inequality (4.26) we
get the conjunction (ii) of the conditions (4.5). This ends the proof of the
Part II and the proof of the whole theorem.
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Faculty of Applied Physics and Mathematics, Gdansk University of Technology, Narutowicza 11/12, 80-952 Gdansk, Poland
E-mail address: [email protected]
Institute of Applications of Mathematics, Warsaw University of Life Sciences (SGGW), Nowoursynowska 159, 00-757 Warsaw, Poland
E-mail address: [email protected]