Periodic points and subsystems of second-order arithmetic

Annals of Pure and Applied North-Holland

Logic 62 (1993) 51-64

51

Periodic points and subsystems of second-order arithmetic Harvey

Friedman*

Department

of Mathematics,

Stephen Department

Xiaokang

Ohio State University,

G. Simpson* of Mathematics,

Columbus,

OH 43210, USA

*

Pennsylvania

State University,

University

Pennsylvania

State University,

Altoona

Park, PA 16802, USA

Yu” *

Department of Mathematics, 16601, USA

Campus,

Altoona,

PA

Communicated by D. van Dalen Received 30 September 1989 Revised 12 June 1992

Abstract Friedman, arithmetic.

H., S.G. Simpson and X. Yu, Periodic points and Annals of Pure and Applied Logic 62 (1993) 51-64.

subsystems

of second-order

We study the formalization within subsystems of second-order arithmetic of theorems concerning periodic points in dynamical systems on the real line. We show that Sharkovsky’s theorem is provable in WKL,. We show that, with an additional assumption, Sharkovsky’s theorem is provable in RCA,,. We show that the existence for all n of n-fold iterates of continuous mappings of the closed unit interval into itself is equivalent to the disjunction of Xi induction and weak Kiinig’s lemma.

1. Introduction

and preliminaries

This paper is a contribution to Reverse Mathematics [9], a program of foundational research whose goal is to classify specific mathematical theorems according to the axioms needed to prove them. In the work presented here, we attempt to determine which axioms in the language of second-order arithmetic are needed to prove Sharkovsky’s Theorem [7] concerning periodic points in dynamical systems on a closed bounded interval of the real line. We do not Correspondence to: Xiaokung Yu, Department of Mathematics, Altoona Campus, Altoona, PA 16601, USA. * Research partially supported by NSF. ** Research partially supported by NSF grant DMS-8701481. 016%0072/93/$06.00

@ 1993 -

Elsevier

Science

Publishers

Pennsylvania

B.V. Ail rights reserved

State

University,

H. Friedman et al

52

completely reach our goal, but we uncover some novel phenomena. This research is therefore presented as an interesting but nondefinitive case study. The reader of this paper will require some knowledge of the techniques by which ordinary mathematics is formalized in subsystems of second-order arithmetic. General information of this kind can be found in [S], [lo], [2], [4], and [ll]. In particular, we assume familiarity with RCA, and WKL,, two of the systems which are most used in Reverse Mathematics. RCA, is the subsystem of second-order arithmetic with A: comprehension and 2: induction. WKL, consists of RCA0 plus weak Kiinig’s lemma, i.e., the axiom that every infinite subtree of 2’” has a path. Here 2
(RC&). A (code for a) complete separable metric space a is to be a pair (A, d) where A is a nonempty countable set and d:A xA+R is a pseudometric, i.e., it satisfies d(a, a) = 0, d(a, b) = d(b, a) 3 0, and d(a, b) + d(b, c) 3 d(a, c). A ( co d e for a) point of a is defined to be a sequence (a,: n E N), a, E A, such that d(a,, a,) G 2-” for all m, n E N with man. Ifx=(a,:nEN) andy=(b,:nEN) arepointsofA, wesetd(x,y)= lim, d(a,, b,), and we define x = y to mean that d(x, y) = 0. The basic open neighborhoods of a are the open balls B(a, r) where a E A, r E Q, r > 0. Here by definition x E B(a, r) if and only if d(x, a) < r. defined

In this paper, the most important examples of complete separable metric spaces are (i) the real line Iw, and (ii) its closed bounded intervals [x, y] where X, y E Iw, x s y. We can make the identification 1w= 0 where Q is the set of rational numbers with the pseudometric d(a, b) = la - bl. Also important are the complete separable metric spaces (iii) 2” of all functions from N into 2 = (0, l}, known as Cantor space, with the countable dense set 2’N and the pseudometric d(f,

g)

=

c

Ifci) ; &)I ,

I

and (iv) NN of all functions from N into N, known as Baire space, with the countable dense set FV” and the pseudometric

A complete separable metric space A is called compact if there eixsts a aniEA, SuchthatforallnEN sequence of finite sequences ((a,;: i ck,):nEN), and b E A there exists i 6 k, such that d(a,i, b) < 2-“. Thus we can prove within RCA0 that the closed bounded interval [x, y] and the Cantor space 2” are compact. Within WKLo, we can prove that compact metric spaces have the Heine-Bore1 covering property, i.e., any covering of d by a countable sequence

Periodic points and subsystems of second-order

of basic open neighborhoods property for compact metric lemma

arithmetic

has a finite subcovering. Indeed, spaces is equivalent over RCA,,

53

the Heine-Bore1 to weak Kiinig’s

(see [l], [2], [ll]).

1.2. Definition

(RCA,,).

(code for a) continuous

If a

and

B are complete

F : a -+ Z?is defined

function

separable

metric

spaces,

a

to be a set of quintuples

Fc_NxAxQ+xBxQ+ which

is required

to have

certain

properties.

Here

Q’

is the

set of positive

numbers. We write rational (a, r)F(b, s) as an abbreviation 3n ((n, a, r, 6, s) E F). Intuitively, (a, r)F(b, s) is a neighborhood condition,

for i.e.,

a piece of information to the effect that F carries B(a, r) into B(b, s). We use the notation (a’, r’) < (a, r) for the inequality d(u, a’) + r’
(a, r)F(b’, s’), then d(b, b’) GS + s’. (a’, r’) < (a, r), then (a’, r’)F(b, s). (b, s) < (b’, s’), then (a, r)F(b’, s’). E > 0, there exist (a, r)F(b, s) such that

d(x, a)
and

s < E. Given F :A-, B as above and x E a, let F(x) be the unique y E g (up to equality in B as defined above) such that d(y, 6) 4s for all (a, r)F(b, s) with d(x, a)
Two continuous

functions

e(x) for all x E a. Given a continuous

function

We write

only

F,, F2 :a *

B are said to be equal

if F,(x) =

F :A -+ I? as above, a modulus of uniform continuity for F is a function h : N + N such that, for all x, y E a, d(x, y) c 2-h(n) implies d(F(x), F(y)) < 2-“. Using the Heine-Bore1 property, we can prove in WKL,, for compact a that every continuous function F : A * I? has a modulus of uniform continuity (see Lemma 2.6 below). A (code for an) open set in a is any set of triples

x E II if and

if 3n 3u 3r ((n, a, r) E I/ A d(x, a) < r).

Thus

the

formula x E U is 27. Conversely, we can prove in RCA0 that for any $’ formula p?(x) there exists an open set U c a consisting of all x E a such that q(x) holds, provided QJ(X) is exfensionul on A in the sense that for all x, y E a, q(x) and x = y imply

q(y)

2. Proof

(see [ll,

011.51).

of Sharkovsky’s

For our purposes, where X is a complete

theorem

in WKLo

a dynamical system is a continuous separable metric space. In dynamical

function systems

F :X-+X, theory, one

H. Friedman et al.

54

studies the behavior of sequences of points obtained by iteration: x, F(x), FZ(x), . . . , F”(x),

...,

where x E X and F”(x) =,FF : - . F(x) , n

for all n E N.

Dynamical systems theory in the context of subsystems of second-order arithmetic has been considered in [l]. A point x E X is said to be periodic of period n if F”(x) = x and F”(x) # x for all m, 1 G m < n. Let ST3 be the following statement, due to Sharkovsky. For any dynamical system F: IR + R on the real line, if there exists a point of period 3, then there exist points of period n for all n > 1, II E N. Our investigations here center on the problem of which axioms of second-order arithmetic are needed to prove ST3. We do not completely solve this problem. In reality, ST3 is only a special case of the much more comprehensive theorem which Sharkovsky actually proved [7], [5], [3]. (The subsequent rediscovery of this special case by Li and Yorke [6] was the impetus for our work here.) Define the Sharkovsky ordering to be the following ordering of the positive integers: 3, 5, . . . ) 2 * 3, 2 * 5, . . . ) 2= . 3, 22 * 5, . . . ) . , . ) 4, 2, 1. The full theorm of Sharkovsky [7] says that m precedes n in the Sharkovsky ordering if and only if every continuous F: R --$ iw with points of period m has points of period n. Our results below concerning ST3 can be generalized so as to apply to the full theorem of Sharkovsky. The proofs require essentially no additional idea beyond Sharkovsky’s combinatorial technique. (For an exposition of Sharkovsky’s proof, see [3] or [5].) With this understanding, we shall concentrate on ST3. We shall now prove a sequence of lemmas leading to the theorem that ST3 is provable in WKL,,. 2.1. Lemma (RCA,,). Let (a,: n E N) and (b,: n E N) be sequences numbers such that a, s a,,, G b,,, such that a, s x s b, for all n.

of rational G b, for all n. Then there exists a real number x

Proof. If there exists a rational number c such that a, G c can take x = c. Otherwise, an argument using _Zyinduction n such that ]a, - b,l < 1. Let n_, be the least such n, and nk<--. by putting Itk+i = the least n such that a, > ck=(u,,+b,,)/2. Then x=(u,,:k~N)=(b,,:k~N)=(c~:k~N)) seen to be a real number with the desired property. 0

< b, for all n, then we shows that there exists define n, < n1 <. . . < ck or b,
The following lemma is an RCA” version of the Intermediate (see also [ll, Chapter II]).

Value Theorem

Periodic points and subsystems of second-order

2.2. Lemma (RCA& I.. F:[x, yl-+R there exists z E [x, y] such that F(Z) = 0. Proof. We imitate number rational

the

standard

arithmetic

55

IScontinuous and F(x) < 0
bisection

argument.

If there

exists

then

a rational

c E [x, y] with F(c) = 0, then we can take z = c. Otherwise, we define numbers a, and b,, n E N, as follows. First, use continuity to find rational

a, and b, such that x < a, < b,, -Cy and F(a,,) < 0 < F(b,).

Given

a,* and b,,, we

have F(c,) f 0, c, = (a, + b,)/2, so let (a,,,, b,+,) = (a,, c,) if F(G) if F(c,)
> 0, (c,, b,) all n2, and z such that

from continuity

0

2.3. Lemma (RCAJ. If F : [0, I]+ R is continuous there exists x E [0, l] such that F(X) =x.

and F[O, 112 [0, 11, then

Proof. Let x0, x1 E [0, 11 be such that F(x,) = 0 and F(x,) = 1. Sestting G(x) = F(x) -x, we have G(x,) < 0 < G(x,) so by Lemma 2.2 there exists x E [x,,, x1] such that G(x) = 0. 0 2.4. Lemma (RC&). x
If U and V are nonempty open sets in R! with U < V (i.e., then lJ
that there exist sequences of rational numbers a, E U, b, E V,
We do not know whether

the following

lemma

is provable

in RCA,,.

2.5. Lemma (WKL,). If F:[a, b]-+ R is continuous and F[a, b] r> [O, 11, then there exist x, y E [a, b] such that F[x, y] = [O, 11. We can also arrange to have either F(x) = 0 and F(y) = 1, or F(x) = 1 and F(y) = 0. Proof. case.)

Assume

F(a) s 0 and

F(b) z 1. (It is easy

to reduce

the proof

to this

H. Friedman et al.

56

We claim

that there

is an open

set U G [a, b] consisting

of all x E [a, b] such

that 3y(x
r\F(y)
(1)

This is in fact provable in RCA”. To see this, note that (1) is equivalent formula, since by continuity we may restrict attention to rational numbers interval Section

[a, b]. The 1 concerning

We claim

existence extensional

that there

of U now

follows

by the

remark

at the

to a 2: y in the end

of

2: formulas.

is an open

set V c [a, b] consisting

of all x E [a, b] such

that Vy(xcysb-+F(y)>O).

(2)

To see this, note that by continuity plus the Heine-Bore1 covering property, (2) is equivalent to the existence of a finite set of neighborhood conditions s,), i s ~1, such that [x, b] is covered by the basic open neighborhoods B(Uj, li)j i =Sn, and B(bi, si) > 0 for all i s n. Thus (2) is equivalent to a _J$’ formula and hence defines an open set. Clearly U < V, so by Lemma 2.4 there exists z such that U < z < V. (If U is empty, we may take z = a.) Since z $ U, we have F(y) 2 0 for all y E [z, b]. Since z $ V, there is x0 with z
2.6. Lemma (WKLJ. Let d and l? be complete separable metric spaces and let F :a --, B be a continuous function. If a is compact, then F has a modulus of uniform continuity. Proof. We first note that property 4 of Definition 1.2 can be strengthened as follows. For all x E a and all E > 0, there exists a neighborhood condition (a, r)F(b,

s) such that d(x, a)
and s < E. To see this, let (a’, r’)F(b,

s) be a

neighborhood condition such that d(x, a’)
Periodic points and subsystems of second-order

With systems

arithmetic

the above lemmas in hand, we are now ready on the real line. In studying a particular dynamical

57

to look at dynamical system F :X+ X, it is

very helpful, indeed essential, to know that the iterations F” :X-+X are defined and make sense for all II E N. Unfortunately, this knowledge is not automatic in subsystems because

of second-order in such

systems

arithmetic it may

with restricted

be nontrivial

induction,

to prove

such as RCA,,

properties

of F” by

induction on n. For continuous F, the whole matter is analyzed completely in Section 4 below. If F is assumed to have a modulus of uniform continuity, our question

is answered

by the following

lemma.

2.7. Lemma (RCAJ. Let X be a complete separable metric space and let F : X + X be a continuous function. If F has a modulus of uniform continuity, then there exist continuous functions F”: X+X, n E N, defined by F’(x) =x, Fn+l(x) = F(F”(x)) for all x E X and n E N.

Proof. Let h’ : IV-+ N be a modulus of uniform continuity for F, and define h: N+ N by h(n) = h ‘(n - 1). We claim that the code of F can be replaced by another code F’ for the same continuous function, with the following property (*); for all n E N and basic open neighborhoods B(a, r) with r < 2-h(n), there exists a neighborhood condition (a, r)F’(b, s) such that s < 2-“. Namely, let F’ be such that (a, r)F’(b, s) holds if and only if there exist n E N and a neighborhood condition (a’, r’)F(b’, s’) such that d(a, a’) < r’ and r < 2-‘@) and s’ < 2Y-’ and (b’, s’ + 2+-l) < (b, s). It is straightforward to check that x E B(a, r) and (a, r)F’(b, s) imply F(x) E B(b, s). Using this, it is easy to see that (*) holds and that F and F’ are codes for the same continuous function. For all n E N, define F” so that (a, r)F”(b, s) holds if and only if there exist neighborhood conditions (a,, rO)F’(al, rJ, (aI, rl)F’(az, r2), . . . , @,-l, r,-@“(a,, r,> with (ao, d = (a, r) and (an, r,J = (b, 3). It is clear that F” has properties 1, 2 and 3 of Definition 1.2. In order to prove property 4, we first use primitive recursion in RCA,, (see [S, $21 of [ll, BII.3]), to obtain the sequence of iterates (h”: n EN), where h”(k) = k and h”+‘(k) = h(h”(k)). We then claim that, for all n, k E N and all basic open neighborhoods B(a, r) with r < 2- h”(k), there exists a neighborhood condition (a, r)F”(b, s) such that s <2-k. To prove the claim, note first that for n = 0 the claim is trivial and for n = 1 it reduces to property (*) above. In general, let n, k E N and O, rO) be given with r0 < 2-h”(k). By Z: induction on m G n, we prove (a, r) = (a that there exist neighborhood conditions (ao, rO)F’(a,, r,), (a,, rl)F’(az, rJ, . . , rm) with r, < 2--h”-m(k). The induction step is given by (*). (a,- ], rm-W(amy Taking m = n, we have our claim. From this, property 4 is immediate. It is clear from the definition of F” that FO(.x) =x and Fn+‘(x) = F(F”(x)) for all n E N and x E X. This completes the proof of Lemma 2.7. 0

58

H. Friedman et al.

Fix a pairing encoding

(Z), In addition

(. , a): N X N -+ N.

function

a sequence

of sets ((Z),:

For any set Z G N, we may regard

Z as

n E N) where

= {i E N: (i, n) E Z}.

we write

(Z)n = {(i, m): (i, m) E Z A m
the finite sequence

is that a strong

Lemma

(WKL,J.

fl

dependent

((Z) m: m < n). The point choice principle

For any n’i formula

is provable

q(n, X, Y)

of the following in WKLo.

in which Z does

not

occur, we have 32 vn VY (q(n, (W, Proof. The proof is in [ll,

Y)*

§VIII.2].

rl(n, (Z)“, (Z),)). q

We are now ready to show that ST3 is provable

in WKLo.

2.9. Theorem (WKLJ. Zf F: R + R is continuous and a point of period 3, then it has points of period n for all n T=1. Proof. Assume that there are three real numbers a < b 3. We first present the informal argument and then indicate the additional considerations which are needed for our formalization within WKL,. Fix n > 3. Informally, we define a sequence of closed bounded intervals

Start with AI = J. Then FAI 2 Z U J so by Lemma 2.5 there exists A, G AI with FA, = J. Then F2A2 2 Z U J so by Lemma 2.5 there exists A3 c A2 with F’A, = J. 2.5 there exists A,_1 LA,_, with . . . Then Fn--2An_2~ Z U J so by Lemma F”-2A,_1 = J. Then F”-lA,_I 2 Z U J so by Lemma 2.5 there exists A, G A,_1 with F”-‘A, = 1. Then F”A, 2 J 1 A,, so by Lemma 2.3 let x E A, be such that F”(x) = x. Clearly x, F(x),

F’(x),

. . . , F”-*(x)

EJ

Periodic points and subsystems of second-order

arithmetic

59

while F”-‘(x) E I, so F”(n) fx for 1 G m
for all x E [w. This

is given

bounded intervals in Iw. The other major difficulty

by Lemmas

within

WKL,

2.6 and is

to

2.7 applied

carry

out

the

to closed recursive

construction of AZ, A,, . . . , A,_1, keeping in mind that the axioms of WKLo include induction only for Z’y formulas. To overcome this difficulty, we use Lemma follows.

2.8 plus the endpoint information Let q(k, X, Y) be a formula saying

in Lemma 2.5. The details are as that X and Y are (codes for) closed

intervals Ak_l, Ak such that Ak EAT-, c J and Fk-‘Ak = J and Fk-’ {endpoints of Ak} = {endpoints of J}. The clause Fk-‘Ak = J is not n’j as it stands. However, the other clauses plus continuity and Lemma 2.2 imply that Fk-‘Ak = J is equivalent to a I_r clause, namely Vr E Q (r E interior

of Ak +- Fk-‘(r)

E J).

Thus q(k, X, Y) is equivalent to a fl formula, and we can apply Lemma 2.8 obtain the sequence of closed intervals (Ak: 1 < k < n). The needed properties these intervals can then be proved by n(: induction, which is equivalent to induction and hence available in WKL”. The inductive step is provided Lemmas 2.2 and 2.5. This completes the proof.

3. An RCA,

to of .ZT by

0

version of Sharkovsky’s

theorem

In this section, we will concentrate on the case that the functions are continuous with modulus of uniform continuity. We show that, in this case, ST3 can be proved in RCA” instead of WKL,. 3.1. Lemma (RCA,,). Zf F: [a, b]+ continuity and F(a) 6 c < d s F(b), thatc’
F(d’)
R’ 1s continuous with a modulus of uniform then there are (ration&) c’, d’ E [a, b] such

andthereise~OsuchthatF(x)>c+eforallx~[c’,b].

Proof. Let E > 0 be such that c + 3~
2.5, we have

Let V be the open set defined by the 2: formula which says that there exists a finite set of neighborhood conditions (a,, r,)F(b,, so, i in, such that (B(a,, ri): i G n) is an open cover of [x, 61, bi > d and Si < E for i s n. Note that for x E [a, b], x E V implies that Vy (x cy < b -+ F(y) > d - e). Therefore U < V.

H. Friedman et al.

60

On

the

other

hand,

F

because

Vy (x my G b -+ F(y) 2 d), then U
then there

for all x e [c’, b]. For

closed

has

a modulus

x E V. Therefore,

is d’>c’such

that F(d’)
[u, U] and

[s, t], we write

of

uniform

continuity,

if

if we let c’ E [a, b] such that Also,

F(x)~c+~E>c+E

q

intervals

[u, V] =)int [s, t] to mean

that

u
E) for some E > 0. It can be arranged to have min{F(s’),

F(t’)}


F(t’)}.

Proof. This lemma is a modified form of Lemma 2.5, proved in RCA” instead of WKLo. In Lemma 2.5, we were looking for one subinterval of the domain which has the precise image we want. In this lemma, we start from two intervals instead of one. The image we want is ‘sandwiched’ between them. In turn, we are looking for two subintervals in the domain whose images are estimates of what we want. Assume Flu and F(b)Sv. From Lemma 3.1, as F(a)~uOsuch Lemma 3.1 again on [s’, b]. As that F(x) > u + e1 for x E [u’, b]. Apply F(b) 2 v > t > F(s’), there are u’, t’ E [s’, b] such that b > v’ > t’ >s’, F(t’) > t, andthereisc2>Osuchthat,F(x)
Theorem

(RCA,).

ST3

holds

for

functions

with modulus

of

uniform

continuity. Proof. Suppose F is a function with a modulus of uniform continuity and F has a point of period 3. The proofs for points of periods 1 and 2 are easy. Now let n E N be a number greater than 1. We want to prove that F has a point of period n + 2. We can assume that there are a < b < c such that F(a) = b, F2(a) = F(b) = c, and F3(a) = F(c) = a. The idea of proving the theorem without weak K&rig’s lemma is to find a sequence of descending intervals (Ak: 1 G k =Sn) such that [b, c] c Fk(A,) c (b*, =J) for 16 k s n, where b* is slightly less than b. We then find A* c A, such that F”+‘(A*) jumps to the left side of b*. We also require that A* c F”+2(A*). If is a fixed point of FnC2, note that Fk(z) > b* for 1 G k c n but ZEA*

Periodic

points and subsystems

of second-order

arithmetic

61

F”+l(Z) -=c b *, then z must be a point of period IZ + 2 for the function. To show this in RCA,,, we first have to choose b* E (a, 6). Note that F(b) = c > b and F’(b) < b. If b* is chosen being close enough to b, we can easily have both F(b*) > b and F2(b *) < b*. For the first intervals, we want A, = [sI, tl] CA: = [u,, u,] such that [b, c] c (F(t,), c] c F(A,) c F(A:) c (b* + 6, 00) f or some 6 >O. Let s1 = b. Apply Lemma 3.1 on [b, F(b*)]. Since F(b)> b > b* >F(F(b*)), there are tl, v1 E [b, F(b*)] such that t1 < ul, F(t,) < b and there is 6 > 0 such that F(x) > b* + 6 for all x E [b, u,]. It is obvious that [b, c] c (F(t,), c] c F(A,). Noticing that F(b) = c, we choose u1 E (8’(tl), b) which is close enough to b such that F[u,, ul] will not fall below b* + 6. Let q(m) be the following statement for m 3 2: There is a sequence quadruples of rationals ( ( uk, Sk, tk, uk): 2 < k C m) such that for 2 G k 5s m, (i) Sk-r < uk 0 such that F[uk, uk] C (&__l + It If

max{F(d, (m + 1)th s,
-

and E).

that v?(m) is a 27 formula since F has a modulus. F(t,)} < CL,
is easy

p(m)

F(tk)); E, u&l

of

to

holds,

see

or

end-points all rt E N. Let Ak = [sk, tk] c Ak+ = [uk, vk] for k C rz. It is obvious that A, cAk for l=sksn. For any k=2,.. ., n , Fk-‘(&l) c Fk(Ak) c Fk(A;) c Fk-‘(A:_,) holds because Ak_, c F(A,) c F(A:) CA&~. We first claim that [b, c] c F”(A,). By using 2’: induction on the statement “b E int Fk(Ak)” for 1 Sk C II, we can prove that b E Fn(A,). Similarly, by using .ZT induction on “b E Int Fk-‘(Ak)” for 2~ k 0 such that Fk(Al) c (b* + E, 00)” is ,Zy as F has a modulus of uniform continuity. Now we find a* E (a, b*) such that F(a*) = tl. The existence follows from the fact that F(a) = b < tl -=cF(b*). Since [a, b*] c F[b, c) c Fn+l(A,), using Lemma 3.1 on the function F”+l, a function with a modulus, we can find an interval A* CA, for the function F”+l such that [a, a*] c F”+l(A*) c (-00, b*). Since A* cAl = [sl, tl] = [b, F(u*)] c F[u, a*], we must have A* c F”+2(A*). It follows from Lemma 2.3 that there is some z E A* which is a fixed point for F”+‘. For z is a point of period k S n, as z E Ak, Fk(z) > b*, but Fn+‘(z) < b*. Therefore, IZ + 2 for the function F. This completes the proof. 0

H. Friedman et al.

62

4. A strange disjunction

Given a dynamical system F: X + X where X is a complete separable metric space, what axioms of second-order arithmetic are needed to prove that F”(x) is defined for all x E X and II E N? The question is natural since, in the absence of this condition, the concepts of dynamical systems theory do not make much sense. We show here that, for arbitrary complete separable metric spaces, ,YSz induction is necessary and sufficient. For compact metric spaces, we show that the disjunction of Cz induction and weak Konig’s lemma is necessary and sufficient. 4.1. Lemma

(RCAJ.

&’ induction is equivalent to flz induction.

Proof. We prove that 2; induction implies z induction. (The proof of the converse is similar.) Let q(n) be a II” formula and assume q(O) and Vi (q(i)+ ~(i + 1)). If there exists n such that q(n) does not hold, fix such an n and write q(i) = (i s n ---, -q(n - i)). Then q(i) is equivalent to a Z!: formula. We have ~(0) and Vi (q(i)* v(i + l)), hence by _Zyinduction q(n), i.e. --v(O), a contradiction. q Ec(: induction implies the following. For all n E N and all complete separable metric spaces X and continuous functions F from X into X, F” is a continuous function from X into X.

4.2. Lemma (RCA,).

Proof. Assume _E’iinduction. By the previous lemma we have @ induction. The code of F” is defined in such a way that (a, r)F”(b, s) holds if and only if there exist neighborhood conditions (a,, rJF(a,, r,), . . . , (a,_,, r,,_I)F(a,, r,,) such that (a, r) = (aO, rO) and (u,, r,,) = (b, s). The only difficulty is in showing that F”(x) is defined for all x E X. For fixed x, we know that F”(x) is defined if and only if for all rational E > 0 there exists a neighborhood condition (a, r)F”(b, s) such that d(a, x)
The following are equivalent:

and all complete separable metric spaces X and continuous into X, Fk is a continuous function from X into X. and all continuous functions F from NN into NN, Fk is a from NN into NN. (Here NN is the Baire space.)

Proof. By Lemmas 4.1 and 4.2, it suffices to prove that statement 3 implies fi induction. Let q(k) be a @ formula and assume ~(0) and Vk (q(k)--, rp(k + 1)). Write q(k) = Vm 3n 8(k, m, n) where 8 is Ei. Put g”(O) = 0 and g,(m + 1) =

Periodic points and subsystems of second-order

arithmetic

63

least rz such that 0(0, m, n). Then go E @ in view of q(O). We now continuous function F from NN into NW. Given g E NN, define F(g) follows. First, put k =g(O) and F(g)(O) = k + 1. Next, given m E N, let least that B(k + 1, m, j) v -0(k, j, g(j + 1)). Such a j must exist in

define a E NN as j be the view of

q(k + 1) v --q(k). If B(k + 1, m, j) holds, put F(g)(m + 1) = j. Otherwise put F(g)(m + 1) = 0. This completes the definition of F. It is straightforward to verify that F is a continuous function from N’ into N’. Applying statement that for all k, g, = Fk(g,) exists and g, E @. We now claim that, gk(0) = k and Vm (g,(m easily proved it follows

by fl

from

3, we see for all k,

+ 1) = the least n such that 8(k, m, n)). This flj

induction.

Z’y induction,

(Note

that rr’: induction

as in Lemma

Vk Vm 3n B(k, m, n), i.e. Vk q(k).

Our theorem

4.1.)

is provable From

the

is now proved.

claim is

in RCA0 since claim,

we have

0

4.4. Lemma (RCA,,). Weak K&zig’s lemma is equivalent to the assertion that 2N is not homeomorphic to NN. (Here 2N is the Cantor space.) Proof. Under weak Kiinig’s lemma, 2’ is not homeomorphic to Nrm, because the former has the Heine-Bore1 covering property while the latter does not. Assume now that weak K&rig’s lemma fails. Let T E 2’N be an infinite tree with no infinite path. As in [ll, §IV.l], let f be the set of T ~2~~ such that t 4 T A Vo (a 5 r+ (T E T). Then for every g E 2N there exists a unique n E N such that {g(O), . . , g(n)) E i? Given g E 2N, define an increasing sequence

n,, = - 1 and ni+i = the 2), . . . , g(n)) E T. Putting

where

least

n >ni

F(g) = ( (g(nj + I), g(nj + 2), . . . , g(ni+i>):

such

that

(g(n, + l), g(n, +

i E N>

we see that F(g) E TN. It is straightforward to verify that F is a homeomorphism of 2N onto TN. Moreover, since p is infinite, p:” is homeomorphic to Nrm. Hence 2N is homeomorphic to NNl. This completes the proof. 0 4.5. 1. 2. from 3. 4.

Theorem (RCA,,). The following are equivalent: The disjunction of Z’z induction and weak K&zig’s lemma. For all k E N and all compact metric spaces X and continuous functions X into X, Fk is a continuous function from X into X. Same as 2 restricted to the compact metric space 2N. Same as 2 restricted to the compact metric space [0, 11.

F

Proof. By Lemma 4.2, 2.6 and 2.7 we see that 1 implies 2. Trivially 2 implies 3 and 4. Suppose now that 1 fails. Then .Zz induction fails. Hence, by Lemma 4.3, we have a continuous function F: NN”‘-tNN such that, for some k, Fk is not a

64

H. Friedman et al.

continuous function from NN to IV. Moreover, weak Konig’s lemma fails, so by Lemma 4.4 we have a homeomorphism @: 2N+ N”. Putting G = @-’ * F * @, we have a continuous function G: 2N’-, 2N such that Gk is not a continuous function from 2” into 2N. Thus 3 fails, and hence trivially 2 fails. Let us make the standard identification of 2” with the Cantor middle third set C E [0, 11. It is straightforward to extend G: C* C to a continuous function H: [0, l]-, [0, 11. Thus 4 fails. This completes the proof. q

References [l] A. Blass, J.L. Hirst and S.G. Simpson, Logical analysis of some theorems of combinatorics and topological dynamics, in: Logic and Combinatorics, Contemporary Math. 65 (Amer. Math. Sot., Providence, RI, 1987) 125-156. [2] D.K. Brown and S.G. Simpson, Which set existence axioms are needed to prove the separable Hahn-Banach Theorem?, Ann. Pure Appl. Logic 31 (1986) 123-144. [3] R.L. Devaney, An Introduction to Chaotic Dynamical Systems (Addison-Wesley, Menlo Park, CA, 1985). [4] H. Friedman, S.G. Simpson and R.L. Smith, Countable algebra and set existence axioms, Ann. Pure Appl. Logic 25 (1983) 141-181; Addendum 28 (1985) 320-321. [5] C.-W. Ho and C. Morris. A graph theoretic proof of Sharkovsky’s theorem on the periodic points of continuous functions, Pacific J. Math. 96 (1981) 361-370. [6] T.-Y. Li and J.A. Yorke, Period three implies chaos, Amer. Math. Monthly 82 (1975) 985-992. [7] A.N. Sharkovsky, Co-existence of the cycles of a continuous mapping of the line into itself, Ukranian Math. Z. 16 (1964) 61-71 (in Russian). [S] S.G. Simpson, Which set existence axioms are needed to prove the Cauchy/Peano theorem of ordinary differential equations?, J. Symbolic Logic 49 (1984) 783-802. [9] S.G. Simpson, Subsytems of Z, and Reverse Mathematics, appendix to: G. Takeuti, Proof Theory, second edition (North-Holland, Amsterdam, 1986) 434-448. lo] S.G. Simpson, Partial realizations of Hilbert’s Program, J. Symbolic Logic 53 (1988) 349-363. 111 S.G. Simpson, Subsystems of Second Order Arithmic, in preparation.