proceedings of the
american mathematical
society
Volume 80, Number 3, November 1980
PERIODIC AND LIMIT ORBITS
AND THE DEPTH OF THE CENTER
FOR PIECEWISE MONOTONE INTERVAL MAPS
ZBIGN1EW NITECKI
Abstract. For a piecewise monotone map of the interval: (a) the non wandering
points outside the closure of the periodic points are isolated in the nonwandering
set; (b) the forward orbit of any such point misses all turning points; (c) the depth
of the center is at most 2; and (d) all w-limit points belong to the closure of the
periodic points.
Let /:/—»/
denote a piecewise-monotone map of the interval: / is continuous,
and strictly increasing or decreasing on each interval of a finite partition / =
[Cq,c,] u • • • l)[ck_x, ck]. The points ci (i = 1, . . ., k — 1) forming the coarsest
such partition are the turning points of /. Denote the nonwandering set of / by ß
and the set of periodic points by P. In general, ß =£P [B], [Y], although generically
ß = P [Y]. However, P always contains the recurrent (Poisson stable) orbits [Y],
even when/is just continuous [CH]. An analysis of the points in ß — P is carried
out in [Y]; in [CH] it is shown (for any continuous/) that ß — P is nowhere dense
in /. In this note, we sharpen the analysis in [Y] to prove the following
Theorem. Iff: I —>I is piecewise monotone, then
(a) each point of ß — P is isolated in ß;
(b) if x E ß - P, thenf(x)
is not a turning point of f, for any n > 0;
(c) ß(/|ß)
= P;
(d) for any z E I, w(z) c P (w = u-limit set).
Statement (b) for n = 0 is shown in [Y]. To put statement (c) in perspective, we
recall the definition of the Birkhoff center of a dynamical system: let ß° be the full
phase space (/ in our case); define ß'+1 = ß(/|ß'), and for limit ordinals define
ß1 = D {&\j < 0- For some ordinal 8, ßÄ = ßÄ+1 = ■ • • ; this set is called the
Birkhoff center, and the least such ordinal 8 is called the depth of the center. It has
been shown that 8 < 2 for flows on orientable compact surfaces [ST] and 8 < 3 on
nonorientable ones [T], [Nl]; on the other hand, 8 is arbitrary for flows on
manifolds of higher dimension and on nonorientable open surfaces [N2] (see
references there to earlier work of Maïer). Since P c ß' for all i, our statement (c)
is the equivalent of 8 < 2 for piecewise monotone interval maps.
All the statements of the theorem are easy corollaries of the following technical
Received by the editors October 18, 1979 and, in revised form, December 10, 1979.
AMS (MOS) subject classifications(1970). Primary 54H20; Secondary 58F22.
© 1980 American Mathematical
0002-9939/80/0000-0578/$02.00
511
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Society
512
ZBIGNIEW NITECKI
Proposition.
If f: I —>I is a piecewise montone map and x £ ß — P, then for
some neighborhood of x, J = J_\J J+, where J± are closed intervals abutting at x,
and
(i)/V+ n / - 0far all n>0;
(ii)fJ_n
J CJ+ for all n>0.
I would like to thank Lai-Sang Young and Ethan Coven for helpful conversations concerning these results, and my colleagues at Northwestern University,
where I began this work on sabbatical, for their hospitality and support.
Proof of Theorem. To see that the proposition implies (a), (c), and (d) of the
theorem, pick x G ß - P: we note that (a) int J+ and int /_ are both wandering
intervals, so ß n int/ = {x}, (c) since x does not return to / on int J is
wandering for/|ß, and (d) any particular orbit enters / at most twice (once in J'_,
once in /+) so that J intersects no «-limit sets.
To see (b), note that if f(x) is a turning point for/, then/n+1 folds some interval
(x — e, x + e), which we can assume is contained in /; to every y £ (x — e, x)
there corresponds z £(x,x
+ e) such that /n+*( v) = fn+k(z) for all k > 0. But
y £ J_ iff z G /+ ; since points of /_ arbitrarily near x return to (x — e, x + e)
under arbitrarily high iterates of /, we have f"+kJ+ n /"+*/_n J ¥= 0 for some
large k, contradicting (i) of the proposition. □
Remark. The following corollaries of the intermediate value property for the real
continuous functions Fn(x) = f(x) — x will be useful in the proof of the proposition:
(1) If K c / is a closed interval and either fn(K) c K or f"(K) D K, then K
contains a fixed point of/", hence K f) P ¥= 0.
(2) If K c / is an interval disjoint from P and f(y)
some v G K, then the same holds for all v G K.
> y (resp. f(y)
< y) for
Proof of Proposition. Pick x G ß — P, and let J = [x — e, x + e] be a closed
neighborhood of x, disjoint from P:
(A) J n P = 0.
Since x is not recurrent (by results mentioned in the introduction), we can adjust e
so that
(B)/"x £ J for any n > 0.
Lemma 1 of [Y] says that
(C) There exist yk —»x and nk —»oo so that f"*yk = x.
A subsequence of the yk belongs to one of the half-intervals [x - e, x] and
[x, x + c]: we denote this one by J_, and let /+ be the other half of /. We can
assume without loss of generality that J_= [x — e, x]. By adjusting e, we can
assume
(D)fN(x - e) = x, andfy ¥=xforn < N and y G int J_.
Our first observation is
(E) Lemma. There exists 8, 0 < S < e, so that f"z £ (x — 8, x) for any n > 0 and
z £J+.
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513
PERIODIC AND LIMIT ORBITS
Proof of (E). For the moment, pick 0 < 8 < e, and suppose for some z E J+
that f'z E (x — 8, x). Then, since f'z < z, (A) implies that f'x < x and hence by
(B),/x <x - e. Thus,
(El)/'[x, z] D [x - e, x - 8].
Now, for e > 0 fixed and 8 > 0 sufficiently small, one of the points yk in (C)
belongs to (x - e, x - 8). Thus,/^
= x >yk; by (D), nk > N, and sof"*(x - e)
= f*-N(x) > x + e by (A) and (B). Hence,
(E2)f"*[x- e,yk]D[x,x
+ e].
But then (El) and (E2) imply
f'+n*[x,z]
Df*[x-e,x-8]
Df*[x-
e,yk] d[x,x
+ e] d[x,z]
contradicting (A). □
Note that by remark (1) above, if f(z) = x < z, then/n(x)
(E). Thus replacing e with 8, we can strengthen (E) to
< x, contradicting
(Ë)/V+ n /_= 0 for all n > 0.
To continue, we invoke piecewise monotonicity. Recall the following notation
from [Y]:
T+(y) = clos{/"y|n > 0),
T_(y) = {zfiyk -» z, nk -* oo with/'V* = v}.
Since there are finitely many turning points for / we can, by shrinking J further,
assume
(F) If c is a turning point off whose forward and backward orbits both hit J, then
x g r+(c) n r_(c).
By [Y], nonwandering turning points belong to P. Thus, by more shrinking, we
can assume J contains no turning points for/.
Next, we show that
(G) IffJ
n J is nonempty, it is an interval, with one endpoint x ± e and the other
either x or /*(c), where 0 < k <n and c is a turning point as in (F).
Proof of (G). If neither endpoint of fj n / is x ± e, then/V c /, contradicting (A). On the other hand, x ± e cannot both be endpoints of fJ, since /"/ = J
would also contradict (A). By (E) and remark (1), no image of x + e can be interior
to J; by (B) and (D), the only image of x — e interior to / is x = fN(x — e).
Finally, if an endpoint of/"/ n / is f(y), y E int J, then y is a turning point off",
so that some turning point c for / satisfies c = f~k(y),
f(y)
= fk(c), k > 0,
n - k > 0. □
(H) The left endpoint offj
n J is not interior to J_.
Proof of (H). If/V n / = [d, x + e] and x - e < d < x, then d = fk(c) for c
some turning point, as in (F). Let (a, b) be a maximal interval containing c such
that/*(a, b) c int /. Unless it is an endpoint of the ambient interval /, each of the
points a and b maps to x + e; since/*/ ÇLJ, at least one of the points a and b is
not an endpoint of L
Since x G T_(c) n (d, x + t), there exist y G fj n / and / such that fy=*c
and hence f'+ky = d <y. On the other hand, since f'+kx G J, some point z
between x and v maps under /' to a (or ¿>-in any case, not an endpoint of /) and
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514
ZBIGNIEW NITECKI
hence f'+k(z) = x + e > z. But then f'+k[d, x + e] D [d, x + e], contradicting
(A). D
A further refinement of (G) and (H) is
(I) For each n > 0, fJ
C\ J (if nonempty) is one of two types: [x — e, d_] with
d_< x, or [d+, x + e] with d+ > x.
Proof of (I). We know that in the second case d+ > x or d+ = x — e. But
d+ = x - e means /"/ D J, contradicting (A). In the first case, if d_> x then
J_£f"J
n /_; but since f"J+ r\ J_= 0, we must have J'_£ f"J'_ , again contradicting (A). □
Now, consider the finite set of turning points c,, . . . , c, for/such that x G Y_(c)
n Y+(c), and for each i, let d¡ be the first forward image of c, in J. Let d0 = x — e,
and set x — 5 = ma\{d¡ < x}.
(J) Lemma.f"J n (x - 8, x) = 0 for all n > 0.
Proof of (J). Pick n the least positive integer for which fJ
intersects (x — 8, x).
By (I), f"J n J must have the form [x - e, d_], with x - 8 < d_< x. By (G),
d_= fk(c¡) for some k < n and c, one of the turning points above; furthermore,
c, G f"~kJ. Now, if d_=t d¡, then d_= f'd¡ for some 0 < / < k, and since d¡ £ J, this
means/'y n (x — 5, x) ¥= 0 with 0 < / < n, contradicting the minimality of n. But
then d_ = d¡, this time contradicting the choice of x — 8. This establishes (J). □
The proposition follows easily: by further shrinking J, we can assume J_C (x —
8, x] so that fJ n int7„= 0 (in other words^/V c /+) for all n > 0. On the
other hand, fJ+ c\ J _= 0 for all n > 0 by (E). These two statements give the
proposition.
References
[B] L. Block, Diffeomorphisms obtained from endomorphisms. Trans. Amer. Math. Soc. 214 (1975),
403-413.
[CH] E. M. Coven and G. A. Hedlund, P - R for maps of the interval, Proc. Amer. Math. Soc. 79
(1980),316-318.
[Nl] D. A. Neumann, Central sequences inflows on 2-manifolds of finite genus, Proc. Amer. Math. Soc.
61 (1976),39-43.
[N2]_,
Central sequences in dynamical systems, Amer. J. Math. 100 (1978), 1-18.
[ST] A. J. Schwartz and E. S. Thomas, 77je depth of the center of 2-mamfolas, Proc. Sympos. Pure
Math., vol. 14,Amer. Math. Soc., Providence, R. I., 1970,pp. 253-264.
[T] E. S. Thomas, Flows on nonorientable 2-manifolds, J. Differential Equations 7 (1970), 448-453.
[Y] L.-S. Young, A closing lemma on the interval, Invent. Math. 54 (1979), 179-187.
Department of Mathematics, Tufts University, Medford, Massachusetts
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