PROCEEDINGS of the AMERICAN MATHEMATICAL SOCIETY Volume 116, Number 3, November 1992 CONTINUOUS FUNCTIONS ON THE INTERVAL THAT HAVE CONJUGATES NEAR THE IDENTITY SAM W. YOUNG (Communicated by James E. West) Abstract. This paper concerns continuous functions on the unit interval and provides solutions to two problems: Characterize those continuous functions that are topologically equivalent (resp. conjugate) to functions arbitrarily near the identity. The second question was raised by Joe Martin. 1. Introduction The purpose of this paper is to establish characterizations in simple terms of continuous functions that respectively have the following two properties. The letter "T" associates with "topologically equivalent" and "C" with "conjugate." A continuous function f: [0, 1] -» [0, 1] has Property T iff for each e > 0 there exist homeomorphisms k , A: [0, 1] -» [0, 1] such that \kfh(x) —x\ < e for all x e [0, 1] (or eq. d(kfh, id) < e .) A continuous function /: [0, 1] -» [0, 1] has Property C iff for each e > 0 there exists a homeomorphism A: [0, 1] -» [0, 1] such that \h~lfh(x)-x\ < e for all x e[0, 1]. The characterization of property C answers a question raised by Joe Mar- tin at the Joint Summer Research Conference, Humbolt State University, June 1989 [CTDS, problem 1.2] and stated again at the Spring Topology Confer- ence, Southwest Texas State University, San Marcos, Texas, April, 1990. The arguments in this paper employ nothing more than the elementary properties of continuous functions on the interval. We will find it convenient to abuse the interval notation by allowing [a, b] = [b, a] = the smallest closed interval containing {a, b). If f is a continuous function defined on an interval and f(a) = f(b), then [a, b] is called a level set interval (of f). And for any x in the domain of /, [x, f(x)] is called a displacement interval (of f). No- tice that every level set interval is covered by the union of two displacement Received by the editors October 10, 1990; presented to the seventh Auburn Miniconference on Real Analysis, October, 1990, sponsored by N.S.F. 1991 Mathematics Subject Classification. Primary 54C05; Secondary 54H15. Key words and phrases. Continuous function, topological equivalence, conjugate, homeomor- phism, homotopy. ©1992 American Mathematical Society 0002-9939/92 $1.00+ $.25 per page 8.33 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 834 S. W. YOUNG intervals; since if f(a) = f(b) = c, then [a, b] = [a, c] U [b, c]. The sym- bol "-»" indicates that the function is onto. The linear orientation reversing homeomorphism on [0, 1] will be denoted by J ; J(x) = 1 - x . 2. Property T Suppose that a < b and c < d . For x e [a, b], let x' = c + (d - c)(x - a)/(b - a). In this section we begin to solve the first problem by establishing Theorem 1. Suppose a < b, c < d, f:[a,b]-»[c,d] is continuous, f~l(c) — {a}, and e > 0. Then there exist homeomorphisms A: [a, b] -» [a, b] and k : [c, d] -» [c, d] such that for all x e [a, b], \kfh(x) - x'\ < s. Lemma. Suppose a < b, c < d, f:[a,b]^»[c,d] is continuous, and f~l(c) = {a}, then there exists a sequence b —xo > x\ > Xi >■■•—►a with the property that /[x,, x0] c (x2, x0] and f[xi+i, x¡] C (x'i+2, x;'_,) for 1 = 1,2,3,.... Proof of Lemma. We begin with xo = b (thus x'0 = d) and choose x\ so that a < Xi < min/_l(x¿). Next choose x2 so that x2 < min/ïxi, xo] and so that a < x2 < min{a + ¿, X\, min/~1(x'1)}. Now by induction we can choose for all i > 3, x, so that x- < min/[x,_i, x¡-2] and so that a < x, < min{c2 + } , x,_i, min/_1(x;'_,)}. The first part of the conclusion is clear since min/[xi,Xo] > x2 and max/[xi,xn] < d — x'0. For i > 1 , we have min/[x,+i, x,] > x'i+2. Now if max/[xí+i, x¡] > x-_, , then there exists t G [x,+i, x¡] such that f(t) = x('_, . But x, < min/~'(x'¡_,) and so max/[x/+i , x,] < x\_x . Thus f[x¡+\, x¡] c (x'j+2, x('_[) and {x,} and {x'¡} are decreasing sequences con- verging to a and c respectively. Proof of Theorem 1. Let 0 < r < 1 be such that (d - c)(\ - r) < e/2. Define A to be an increasing homeomorphism on [a, b] such that h(a + (b —a)r') —x, for i — 0, 1, 2, ... Define Acto be an increasing homeomorphism on [c, d] such that k(x[) = (a + (b - a)r')' for i - 0, 1,2, ... The sequences {x,} and {x;'} are given by the lemma. First consider x e [a+(b-a)r, b]. Then A(x) e [xi, x0], /A(x) e (x2, x0], and kfh(x) e ((a+(b-a)r2)', b]. Now suppose i > 1 and x e [a + (b-a)r'+i , a + (b - a)rl]. Then A(x) € [x,+i, x,], /A(x) e (x('+2, x,'_,), and kfh(x) e ((a + (b - a)ri+2y , (a +(b - a)r1-1)'). Of course kfh(a) = f(a) = c. We see that where ever x is located in the partition of [a, b] provided by the sequence {a + (b - a)r'}, the image kfh(x) and x' are located in the same or in adjacent intervals of the partition of [c, d] provided by the sequence {(a + (b- a)r')'}. Thus the distance from kfh(x) to x' in less than twice the length of the longest interval, which is [(a + (b-a)r)', d]. So for all x 6 \a, b], \kfh(x) - x'| < 2(d - c)(l - r) < e . Notice that in the special case a = c and b — d, we have x' — x, the homeomorphism Ac can be chosen to be A-1 , and |A_1/A(x) - x| < e for all x e [a, b]. And thus we have License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use continuous functions on the interval 835 Corollary. Suppose a < b, f: [a, b] -» [a, b] is continuous, f~x(a) = {a}, and e > 0. Then there exists a homeomorphism A: [a, b] -* [a, b] such that for all x e [a, b], \h~lfh(x) - x| < e . Theorem 2. Suppose f: [0, 1] -» [0, 1] is continuous, then the following are equivalent: Ti. If e > 0, /Aere ex/sí homeomorphisms Ac, A: [0, 1] -» [0, 1] swcA /Aa/ \kfh(x) - x| < e ./br a// x e [0, 1] (property T). T2. No finite collection of level set intervals of f covers [0, 1]. T3. There exists a function g e {f,fJ, Jf, JfJ} and 0 < a < 1 swcA /Aai g(s) < g(«) < g(t) for all 0<s <a<t < 1. Property T3 is described this way in order to avoid dealing with four cases of inequalities. If g - f, for example, we see that the graph of / is inscribed in two rectangular boxes (the left one is possibly degenerate) that meet at the point (a, f(a)). The graph of / is allowed to touch the top of the left box but not the bottom of the right box except at (a, f(a)). When we choose g = fJ , Jf, or JfJ, the graph is flipped to the different orientations but still has the special point with respect to the two boxes. T2 => T3. We begin with the observation that if /~'(0) is connected and contains 0, then we can choose g = f and a — max/_1(0) to satisfy T3. Likewise, if f~x(0) is connected and contains 1, then choose g = fJ and a = max(//)~'(0) = maxJ/_1(0). We see that in any case where e is an end point of [0, 1] and f~l(e) is connected and contains an end point of [0, 1], we only have to choose the appropriate g and a to obtain condition T3. Thus we can move on to the case where c - max/_1(0) > 0, d = min/~'(l) < 1, and / is not constant on either [0, c] or [d, 1]. And we do not lose generality by assuming that c < d since otherwise, if d < c, we could choose to work with the function fJ and have fJ(J(c)) = f(c) = 0, fJ(J(d)) = f(d) = 1, and J(c) < J(d). Let G be the collection of all open intervals (s, t), s / / such that f(s) = f(t), or in other words, the interiors of nondegenerate level set intervals of /. In the case under consideration, we must have that some element of G contains c. This is because f is not identically 0 on [0, c] so that a number s, 0 < s < c can be chosen so that f(s) > 0 and a number t, c < t < 1 so that /(/) = f(s). Similarly, d belongs to an element of G. By the same sort of argument, either [0, c] is a level set interval or a subset of an element of G. In light of the above, it follows from the covering theorem that [c, d] is not covered by G and so there exists a number a , c < a < d such that a belongs to no element of G.