1
Acta Math. Univ. Comenianae
Vol. (submitted to AMUC), pp. 1–11
NON-MONOTONICITY HEIGHT
OF PM FUNCTIONS ON INTERVAL
PINGPING ZHANG
es
s
Abstract. Using the piecewise monotone property, we give a full description of
non-monotonicity height of PM functions with a single fort on compact interval.
This method is also available for general PM functions with finitely many forts, as
well as those functions defined on the whole real line. Finally, we obtain a sufficient
and necessary condition for the finite non-monotonicity height by characteristic
interval.
1. Introduction
Pr
Let I := [a, b] be a closed interval and f : I → I be a continuous function. A
point c ∈ (a, b) is called a fort if f is strictly monotonic in no neighborhood of
c. As in [11], a continuous function f : I → I is said to be piecewise monotonic
(abbreviated as PM function) if the number N (f ) of forts is finite. Then the
sequence N (f n ) satisfies the ascending relation
0 = N (f 0 ) ≤ N (f ) ≤ · · · ≤ N (f n ) ≤ N (f n+1 ) ≤ . . . .
In
Denote P M (I, I) the set of all PM functions on I and let the least k ∈ N ∪ {0}
satisfying N (f k ) = N (f k+1 ) if such k exists, and ∞ otherwise, be the nonmonotonicity height H(f ) of f (cf. [13, 8]).
It is known that non-monotonicity height H(f ) is closely related to the problem
of iterative roots. In 1961, Kuczma [2] gave a complete description of iterative
roots for PM functions with H(f ) = 0. In 1983, J. Zhang and L. Yang [11]
put forward “characteristic interval K(f )” for PM function, and for the first time
applied it to obtain monotonic iterative roots if those functions satisfy H(f ) =
1. From [11, 13], we know that arbitrary PM functions satisfying H(f ) ≥ 2
have no continuous iterative roots of order greater than N (f ). Later, L. Liu
and W. Zhang [9] proved that every continuous iterative root of a PM function
with H(f ) ≤ 1 is an extension an iterative root of f of the same order on the
characteristic interval K(f ). Further results on iterative roots of PM functions with
H(f ) ≥ 2 appeared in [6, 8]. Non-monotonicity height as assumption condition
also appears in topological conjugacy between PM functions. In 2013, Y. Shi, L.
Received September 5, 2015; revised July 14, 2016.
2000 Mathematics Subject Classification. Primary 39B12, 37E05, 26A18.
Key words and phrases. iteration; fort; non-monotonicity height; piecewise monotonic function.
Supported by ZR2014AL003, J12L59 and 2013Y04 grants.
2
PINGPING ZHANG
Pr
es
s
Li and Z. Leśniak [10] constructed all homeomorphic solutions and continuously
non-monotone solutions of the conjugacy equation ϕ ◦ f = g ◦ ϕ, where f : I →
I, g : J → J are two given r-modal interval maps (special PM functions) with
H(f ) = H(g) = 1 and I, J are closed intervals. For a class of PM functions with
H(f ) ≥ 1, [3] gave a sufficient and necessary condition under which any two of
these maps are topologically conjugate. Very recently, L. Li and W. Zhang [5, 7]
gave a sufficient condition as well as a method of constructing the topological
conjugacy between PM functions f and their iterative roots if H(f ) = 1.
As known from the previous work, H(f ) plays an important role in studying
PM functions. Thus, one problem was raised naturally: How to determine the
H(f ) for a given PM function f ? This question was considered by Lin Li in [4].
It turns out that even very simple PM functions can become quite complicated
under iteration. [4] investigated a class of polygonal functions with a unique vertex
and determined the number of vertices under iteration by analyzing the slope of
polygon. Those polygonal functions with opposite sign of slope are special kinds
of PM functions, among which the particular case f (a) = a and f (b) = b also
appeared in reference [1]. It seems that up to now there has not been a complete
result of this problem (see [12]).
In this paper, we are interested in non-monotonicity height H(f ) of PM functions. In Section 2, we give a full description of H(f ) for those functions with a
single fort on compact interval. Our method using the piecewise monotone property is also available for general PM functions with finitely many forts, as well as
those functions defined on the whole real line. Section 3 illustrates this method for
those general PM functions by several examples. Finally, in Section 4, we discuss
the relations between non-monotonicity height H(f ) and characteristic interval
K(f l ) for a given PM function f .
2. PM functions with a unique fort
In
The main difficulties in finding H(f ) of PM functions come from the sharply
increasing number of forts under iteration. Using the piecewise monotonicity, we
find that N (f n ) depends on the second order iterate of f . Furthermore, we prove
that H(f ) is uniquely determined by the second order iterate of all those critical
points such as fort and two endpoints as well as fixed points of the given PM
function.
In order to determine H(f ) by observing the change of N (f n ) and to avoid
complicated computation, in this section, we only consider PM function on the
compact interval I := [a, b], each of which has only one fort.
2.1. f1 is increasing and f2 is decreasing
Let a function f ∈ P M (I, I) be of the form
(
(2.1)
f (t) =
f1 (t),
a ≤ t ≤ t0 ,
f2 (t),
t0 < t ≤ b,
NON-MONOTONICITY HEIGHT OF PM FUNCTIONS ON INTERVAL
3
where f1 , f2 are continuous, strictly monotonic with different monotonicities. In
this subsection, we investigate that f1 is increasing and f2 is decreasing, the main
result on H(f ) reads as follows.
Theorem 1. Suppose that f ∈ P M (I, I) is of the form (2.1).
(i) If t0 ≥ f (t0 ), then H(f ) = 1 and
lim f n (t0 ) = t∗ ,
(2.2)
n→∞
s
where t∗ is a fixed point of f.
(ii) If min{f (a), f (b)} ≥ t0 , then H(f ) = 1. Moreover, (2.2) holds in the case
where the unique fixed point t∗ of f is attracting.
(2.3)
es
Proof. In case (i), it follows from the condition t0 ≥ f (t0 ) that f (t) ∈ [a, t0 ] for
all t ∈ [a, b], which implies the iterate f n for n ≥ 1, proceeds on the monotonic
subinterval [a, t0 ], and its number of forts is invariant under iteration, i.e., N (f n ) ≡
N (f ) = 1 for all n ≥ 2. Thus H(f ) = 1 is proved. Furthermore, the monotonicity
of f1 yields
f (t0 ) ≥ f 2 (t0 ) ≥ f 3 (t0 ) ≥ . . . f n (t0 ) ≥ · · · ≥ a,
Pr
that is, {f n (t0 )} is a decreasing sequence which tends to t∗ ∈ [a, t0 ). Moreover, by
continuity of f, f (t∗ ) = t∗ and (2.2) is proved.
In case (ii), inequality min{f (a), f (b)} ≥ t0 shows that
f (t) ∈ [min{f (a), f (b)}, f (t0 )] ⊆ [t0 , b],
t ∈ [a, b].
Thus the iterate f n proceeds on the monotonic subinterval [t0 , b], i.e., N (f n ) ≡
N (f ) = 1 for all n ∈ N, which implies that H(f ) = 1. Moreover, we assert that
(2.4) f 2 (t0 ) ≤ f 4 (t0 ) ≤ . . . ≤ f 2n (t0 ) ≤ . . . ≤ t∗ ≤ . . . ≤ f 2n−1 ≤ . . . ≤ f 3 (t0 ) ≤ f (t0 ).
In fact, f ([t0 , b]) ⊆ [t0 , b] yields f n (t0 ) ∈ [t0 , b] for any positive integer n. Since f 2
is strictly increasing on [t0 , b] and t0 ≤ f 2 (t0 ), there exists t∗ ∈ (t0 , b] such that
f 2n (t0 ) < t∗
In
(2.5)
and
f 2n (t0 ) < f 2n+2 (t0 ).
By the same argument, we also have
(2.6)
f 2n−1 (t0 ) > t∗
and
f 2n−1 (t0 ) > f 2n+1 (t0 ).
Then (2.4) is proved by (2.5)–(2.6), which gives the assertion of (ii). This completes
the proof.
Remark 1. Under the assumption of case (ii), the function f can have a point
z of period 2 such that t0 < z < t∗ < f (z) < f (t0 ).
Theorem 2. Suppose that f ∈ P M (I, I) is of the form (2.1).
(i) If min{f (a), f 2 (t0 )} ≥ t0 > f (b), then H(f ) = 2.
(ii) If min{f (b), f 2 (a)} ≥ t0 > f (a), then H(f ) = 2.
(iii) If f (t0 ) > t0 > max{f (a), f (b)} and min{f 2 (t0 ), f 2 (a), f 2 (b)} ≥ t0 , then
H(f ) = 2.
4
PINGPING ZHANG
Proof. In case (i), the assumption f (a) ≥ t0 > f (b) leads to the result


a ≤ t ≤ t0 ,
 f2 ◦ f1 (t),
2
(2.7)
f (t) =
f2 ◦ f2 (t),
t0 < t ≤ f −1 (t0 ),


f1 ◦ f2 (t),
f −1 (t0 ) < t ≤ b,
which implies that t0 , f −1 (t0 ) are two forts of f 2 . Using the assumption
min{f (a), f 2 (t0 )} ≥ t0 ,
s
we obtain that
(2.8)
f 2 (t) ∈ [min{f (a), f 2 (t0 ), f 2 (b)}, f (t0 )] ⊆ [t0 , f (t0 )] ⊆ [t0 , f −1 (t0 )],
−1
t ∈ [a, b].
−1
es
Noticing f ([t0 , f (t0 )]) ⊂ [t0 , f (t0 )] ⊂ [t0 , f (t0 )], it follows that the iterate
f n (n ≥ 3) proceeds on the monotonic subinterval [t0 , f −1 (t0 )]. Thus N (f n ) ≡
2 (n ≥ 2) is a consequence of the monotonicity of f 2 , and then H(f ) = 2.
Regarding case (ii), with a similar proof as that of (i), we have f ([a, b]) ⊆
[f (a), f (t0 )] from the assumption f (a) < t0 ≤ f (b). Then


a ≤ t ≤ f −1 (t0 ),
 f1 ◦ f1 (t),
2
(2.9)
f (t) =
f2 ◦ f1 (t),
f −1 (t0 ) < t ≤ t0 ,


f2 ◦ f2 (t),
t0 < t ≤ b,
Pr
implying f −1 (t0 ), t0 are two forts of f 2 . The formula (2.9) also shows that
max f 2 = max{f1 ◦ f1 (f −1 (t0 )), f2 ◦ f2 (b)} ≤ f (t0 ) ≤ b,
min f 2 = min{f1 ◦ f1 (a), f2 ◦ f1 (t0 )} ≥ t0 ,
that is, f 2 (t) ∈ [t0 , b] for all t ∈ [a, b]. Hence, the iterate f n (n ≥ 3) proceeds on
the monotonic subinterval [t0 , b] and N (f n ) ≡ N (f 2 ) = 2 for all integers n ≥ 2.
Therefore, H(f ) = 2.
In
In case (iii), the assumption f (t0 ) > t0 > max{f (a), f (b)} shows that f ([a, b]) ⊆
[min{f (a), f (b)}, f (t0 )], and then

f1 ◦ f1 (t),
a ≤ t ≤ f1 −1 (t0 ),



 f ◦ f (t),
f1 −1 (t0 ) < t ≤ t0 ,
2
1
(2.10)
f 2 (t) =

f2 ◦ f2 (t),
t0 < t ≤ f2 −1 (t0 ),



f1 ◦ f2 (t),
f2 −1 (t0 ) < t ≤ b,
which follows that f1 −1 (t0 ), t0 , f2 −1 (t0 ) are three forts of f 2 . Now (2.10) jointly
with the assumption t0 ≤ min{f 2 (t0 ), f 2 (a), f 2 (b)}, imply
max f 2 = max{f1 ◦ f1 (f1 −1 (t0 )), f2 ◦ f2 (f2 −1 (t0 ))} = f (t0 ) ≤ f2 −1 (t0 ),
min f 2 = min{f1 ◦ f1 (a), f2 ◦ f1 (t0 ), f1 ◦ f2 (b)} ≥ t0 ,
that is, f 2 ([a, b]) ⊆ [t0 , f2 −1 (t0 )], and the iterate f n (n ≥ 3) proceeds on the
monotone subinterval [t0 , f2 −1 (t0 )]. Thus, N (f n ) ≡ N (f 2 ) = 3 for all n ≥ 2 and
H(f ) = 2. This completes the proof.
NON-MONOTONICITY HEIGHT OF PM FUNCTIONS ON INTERVAL
5
Theorem 3. Suppose that f ∈ P M (I, I) is of the form (2.1).
(i) If f has no fixed points on [a, t0 ) and f (b) ≥ t0 > f 2 (a), then H(f ) ∈ (2, ∞)
is finite.
(ii) If f has no fixed points on [a, t0 ) and f (t0 ) > t0 > max{f (a), f (b)}, f 2 (t0 ) ≥
t0 > min{f 2 (a), f 2 (b)}, then H(f ) ∈ (2, ∞) is finite.
s
Proof. In case (i), we first claim that t0 > f (a). Otherwise, the assumption
f (b) ≥ t0 implies that f (t) ≥ t0 for all t ∈ [a, b]. Then f 2 (t) ≥ t0 is a contradiction
to the condition t0 > f 2 (a). Hence, f ([a, b]) ⊆ [f (a), f (t0 )] and


a ≤ t ≤ f −1 (t0 ),
 f1 ◦ f1 (t),
(2.11)
f 2 (t) =
f2 ◦ f1 (t),
f −1 (t0 ) < t ≤ t0 ,


f2 ◦ f2 (t),
t0 < t ≤ b.
es
We conclude f −1 (t0 ), t0 are two forts of f 2 . Note that f is strictly increasing on
subinterval [a, t0 ] with no fixed point, it is true that f (t) > t for t ∈ [a, t0 ] and
there exists a finite positive integer i such that
f i (a) ≥ t0 > f i−1 (a) > · · · > f (a).
(2.12)
Pr
Moreover, the assumption f (b) ≥ t0 yields f n (t0 ) ≥ t0 for any integer n ≥ 1.
Thus we get f i ([a, t0 ]) ⊆ [t0 , b]. On the other hand, by the facts that f (b) ≥ t0
and f is strictly decreasing on [t0 , b], f is a self-mapping on [t0 , b], and then
f n ((t0 , b]) ⊆ [t0 , b] for any n ∈ N. Hence, f i ([a, b]) ⊆ [t0 , b] and thus the number
of the forts of f n (n ≥ i) is identical to f i , that is, N (f n ) is bounded for n → ∞.
Therefore, H(f ) = A < ∞ for an integer A > 2.
In case (ii), by using the assumptions f 2 (t0 ) ≥ t0 and the monotonicity of f on
subinterval [t0 , b], for an arbitrary integer n ≥ 1 we get
t0 ≤ f 2 (t0 ) ≤ · · · ≤ f 2n (t0 ) ≤ · · · ≤ f 2n−1 (t0 ) ≤ · · · ≤ f 3 (t0 ) ≤ f (t0 ),
In
inductively. The formula (2.12) means that f has a fixed point t∗ ∈ (t0 , f (t0 )).
Note that f has no fixed points and is strictly increasing on subinterval [a, t0 ],
it is true that f (t) > t for all t ∈ [a, t0 ]. Consequently, f 2 (b) > f (b) and
f 2 (a) > a since f (t0 ) > t0 > max{f (a), f (b)}. Hence, min{f 2 (a), f 2 (b)} >
min{f (a), f (b)}. As in case (i), repeating this process, we get a strictly increasing
sequence {min{f n (a), f n (b)}} fulfilling
(2.13)
min{f k (a), f k (b)} ≥ t0
for certain positive integer k. Then by using the results f (t0 ) ≤ f2 −1 (t0 ), (2.12)
and (2.13), we get f k ([a, b]) ⊆ [t0 , f2 −1 (t0 )], and then N (f n ) ≡ N (f k ) for all
integers n ≥ k + 1. Therefore, N (f n ) is finite for n → ∞ and H(f ) = B < ∞ for
an integer B > 2. This completes the proof.
6
PINGPING ZHANG
Theorem 4. Suppose that f ∈ P M (I, I) is of the form (2.1).
(i) If f (a) ≥ t0 > max{f (b), f 2 (t0 )}, then H(f ) = ∞.
(ii) If f (t0 ) > t0 > max{f (a), f (b), f 2 (t0 )}, then H(f ) = ∞.
(iii) If f has fixed points on [a, t0 ) and f (t0 ) > t0 , then H(f ) = ∞.
Proof. In case (i), from the assumption f (a) ≥ t0 > f (b), we have
f ([a, b]) = [f (b), f (t0 )] = [f (b), t0 ] ∪ (t0 , f (t0 )].
s
Then f (f [a, b]) = [f 2 (b), f (t0 )] ∪ [f 2 (t0 ), f (t0 )] ⊃ [f 2 (t0 ), f (t0 )], which covers the
point t0 since f 2 (t0 ) < t0 < f (t0 ). Hence, f 2 has two forts t0 , f2 −1 (t0 ). Following
the same process, one checks that f 3 has at least three forts t0 , f2 −1 (t0 ), f2 −2 (t0 )
and {N (f n )} is strictly increasing for n. Therefore, N (f n ) is unbounded as n →
∞, and then H(f ) = ∞.
es
In case (ii), the assumption f (t0 ) > t0 > max{f (a), f (b)} implies that the
points f1 −1 (t0 ), f2 −1 (t0 ), t0 are forts of f 2 . The remaining proof is similar as
that of (i). That is, by the condition t0 > f 2 (t0 ), every range f n ([a, b]) (n ≥ 2)
covers the interval [f 2 (t0 ), f (t0 )] containing t0 as its interior point. Then new forts
appear after every iteration and limn→∞ N (f n ) = ∞. Hence, H(f ) = ∞.
Pr
In case (iii), without loss of generality, assume that t1 ∈ [a, t0 ) is a fixed point
of f. Note that f (t0 ) > t0 , then as discussed for case (i)-(ii), every f n ([a, b])
covers the interval [t1 , f (t0 )] for any integer n ≥ 1, which contains t0 as its interior point. Therefore, new forts appear continually under every iteration and
limn→∞ N (f n ) = ∞. Therefore, H(f ) = ∞. This completes the proof.
The main result of Theorems 1–4 is presented in Table 1 as follows.
Table 1. Results on nonmonotonicity height H(f ).
In
Conditions
t0 ≥ f (t0 )
t0 < f (t0 ) min{f (a), f (b)} ≥ t0
f (b) ≥ t0 > f (a);
f (t) > t, t ∈ [a, t0 )
f (a) ≥ t0 > f (b)
f 2 (a) ≥ t0
f 2 (a) < t0
f 2 (t0 ) ≥ t0
f 2 (t0 ) < t0
max{f (a), f (b)} < t0 ;
f 2 (t0 ) ≥ t0 min{f 2 (a), f 2 (b)} ≥ t0
f (t) > t, t ∈ [a, t0 )
min{f 2 (a), f 2 (b)} < t0
f 2 (t0 ) < t0
f has fixed points on [a, t0 )
H(f ) = ∞
Nonmonotonicity
height
H(f ) = 1
H(f ) = 1
H(f ) = 2
2 < H(f ) < ∞
H(f ) = 2
H(f ) = ∞
H(f ) = 2
2 < H(f ) < ∞
H(f ) = ∞
2.2. f1 is decreasing and f2 is increasing
In this subsection, we investigate the function (2.1) in which f1 is decreasing and
f2 is increasing. The discussion is similar to that of the Subsection 2.1 since
g(t) := a + b − f (t) for all t ∈ [a, b]. Therefore, we only give the result and omit
their proofs.
NON-MONOTONICITY HEIGHT OF PM FUNCTIONS ON INTERVAL
7
Theorem 5. Suppose that f ∈ P M (I, I) is of the form (2.1).
(i) If f (t0 ) ≥ t0 , then H(f ) = 1 and (2.2) holds, where t∗ is a fixed point of f .
(ii) If t0 ≥ max{f (a), f (b)}, then H(f ) = 1 and (2.2) holds, where t∗ is the
unique fixed point of f .
Theorem 6. Suppose that f ∈ P M (I, I) is of the form (2.1).
(i) If f (b) ≥ t0 > f (a) and t0 ≥ f 2 (b), then H(f ) = 2.
(ii) If f (a) ≥ t0 > f (b) and t0 ≥ f 2 (t0 ), then H(f ) = 2.
(iii) If min{f (a), f (b)} > t0 > f (t0 ) and t0 ≥ max{f 2 (t0 ), f 2 (a), f 2 (b)}, then
H(f ) = 2.
es
s
Theorem 7. Suppose that f ∈ P M (I, I) is of the form (2.1).
(i) If f has no fixed points on (t0 , b] and f 2 (b) > t0 > f (a), then H(f ) ∈ (2, ∞)
is finite.
(ii) If f has no fixed points on (t0 , b] and min{f (a), f (b)} > t0 > f (t0 ),
max{f 2 (a), f 2 (b)} > t0 ≥ f 2 (t0 ), then H(f ) ∈ (2, ∞) is finite.
Theorem 8. Suppose that f ∈ P M (I, I) is of the form (2.1).
(i) If f (a) ≥ t0 > f (b) and f 2 (t0 ) > t0 , then H(f ) = ∞.
(ii) If min{f (a), f (b), f 2 (t0 )} > t0 > f (t0 ), then H(f ) = ∞.
(iii) If f has fixed points on (t0 , b] and t0 > f (t0 ), then H(f ) = ∞.
Pr
Table 2 presents all cases listed in Theorems 5–8 as follows.
Table 2. Results on nonmonotonicity height H(f ).
Conditions
f (t0 ) ≥ t0
f (t0 ) < t0 max{f (a), f (b)} ≤ t0
f (a) < t0 ≤f (b);
f (t) < t, t ∈ [t0 , b)
f (b) < t0 ≤ f (a)
In
f 2 (b) ≤ t0
f 2 (a) > t0
f 2 (t0 ) ≤ t0
f 2 (t0 ) > t0
min{f (a), f (b)} > t0 ;
f 2 (t0 ) ≤ t0 max{f 2 (a), f 2 (b)} ≤ t0
f (t) < t, t ∈ [t0 , b)
max{f 2 (a), f 2 (b)} > t0
f 2 (t0 ) > t0
f has fixed points on [t0 , b)
H(f ) = ∞
Nonmonotonicity
height
H(f ) = 1
H(f ) = 1
H(f ) = 2
2 < H(f ) < ∞
H(f ) = 2
H(f ) = ∞
H(f ) = 2
2 < H(f ) < ∞
H(f ) = ∞
3. PM functions with finitely many forts
From the proofs of Theorems 1–8 and Tables 1–2, we note that H(f ) is independent
of concrete route, and is uniquely determined by the second order iterate of the
unique fort and two endpoints as well as fixed points of the given PM function.
Our method using the piecewise monotone property is also available for general
PM functions with finitely many forts, as well as those functions defined on the
whole real line. In the following, we present several examples illustrating that
8
PINGPING ZHANG
H(f ) of the general PM functions is also determined by the second iterate of its
critical points such as forts, two endpoints and fixed points.
Clearly, φ1 has three forts t1 := 15 , t2 :=
s
Example 1. Consider the mapping φ1 : [0, 1] → [0, 1] defined by
 2
4
1

 − 3 t + 5 , t ∈ [0, 5 ],



 1
3

 3 t + 35 ,
t ∈ ( 51 , 10
],
φ1 (t) :=

3 2

−t + 1,
t ∈ ( 10
, 5 ],





 1
2
t ∈ ( 52 , 1].
2t + 5,
3
10 , t3
:= 25 . Note that
es
min{φ1 (t1 ), φ1 (t3 )} > t3 ,
In
Pr
then H(φ1 ) = 1 by (i) of Theorem 5.
Example 2. Consider the mapping φ2 : (−∞, 4] → (−∞, 4] defined by
 2
t ∈ (−∞, 1],

3 t,





 − 2 t + 4 , t ∈ (1, 2],

3
3
φ2 (t) :=

3

t ∈ (2, 3],

2 t − 3,




 3
− 2 t + 6, t ∈ (3, 4].
It is easy to see that t1 := 1, t2 := 2, t3 := 3 are forts of φ2 . Furthermore,
φ2 (t3 ) > t1 > lim φ2 (x)
x→−∞
and
implying that H(φ2 ) = 2 by (i) of Theorem 6.
t1 > φ22 (t3 ),
9
es
s
NON-MONOTONICITY HEIGHT OF PM FUNCTIONS ON INTERVAL
Pr
Example 3. Consider the mapping φ3 : R → R defined by

4t,
t ∈ (−∞, 1],







 −4t + 8, t ∈ (1, 2],
φ3 (t) :=


t − 2,
t ∈ (2, 3],






4 − t,
∈ (3, +∞).
In
Clearly, t1 := 1,t2 := 2 and t3 := 3 are forts of φ3 . Since limt→∞ φ3 (t) = −∞ and
φ3 (t1 ) > t1 > φ23 (t1 ), we obtain that H(φ3 ) = ∞ similar to (ii) of Theorem 4.
10
PINGPING ZHANG
4. Relations between H(f ) and K(f l )
In this subsection, we give a sufficient and necessary condition under which H(f )
of a PM function f is finite.
Theorem 9. If f ∈ P M (I, I), then H(f ) is finite if and only if there exists a
characteristic interval K(f l ) for an l ∈ N.
(4.1)
es
s
Proof. Sufficiency. If there exists a characteristic interval K(f l ) of the PM
function f l for an l ∈ N, then the iterate (f l )n of f l proceeds on the strictly
monotonic subinterval K(f l ) ⊂ I and the number N (f nl ) of forts is invariant under
iteration, i.e., N (f nl ) ≡ N (f l ) for all n ≥ 1. Thus H(f l ) is finite. Consequently,
H(f ) is finite by using N (f ) ≤ N (f l ) and the finity of H(f l ).
Necessarity. If H(f ) is finite, then N (f H(f )+i ) = N (f H(f ) ) for all i ∈ N.
Consequently, we obtain
N ((f H(f ) )n ) = N (f H(f ) ),
n ∈ N.
Let l = H(f ). The formula (4.1) changes into N (f nl ) = N (f l ), which shows
that the iterate of f l is invariant on forts. According to the definition of characteristic interval, there exists a strictly monotonic subinterval K(f l ) ⊂ I as the
characteristic interval of f l such that f l (I) ⊂ K(f l ).
Pr
Theorem 9 implies an essential difference between H(f ) < ∞ and H(f ) = ∞.
That is, H(f ) < ∞ guarantees a characteristic interval of f l for l ∈ N, which
describes some invariance of f , in turn those properties determine its dynamical
behavior. While for H(f ) = ∞, there is no such a strictly monotonic subinterval
since the range of f n (n = 1, 2, . . . ) covers at least one fort under each iteration.
Acknowledgment. The author is grateful to the referees for their careful
reading and comments.
In
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Pingping Zhang, Binzhou University, Shandong 256603, P. R. China,
e-mail: [email protected]