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Discrete Dynamics in Nature and Society
Volume 2011, Article ID 132985, 7 pages
http://dx.doi.org/10.1155/2011/132985
Research Article

Topological Entropy and Special 𝜶-Limit Points of Graph Maps

College of Mathematics and Information Science, Guangxi University, Nanning, Guangxi 530004, China

Received 11 December 2010; Revised 4 February 2011; Accepted 2 March 2011

Academic Editor: M. De la Sen

Copyright © 2011 Taixiang Sun et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Let 𝐺 a graph and 𝑓𝐺𝐺 be a continuous map. Denote by (𝑓), 𝑅(𝑓), and SA(𝑓) the topological entropy, the set of recurrent points, and the set of special 𝛼-limit points of 𝑓, respectively. In this paper, we show that (𝑓)>0 if and only if SA(𝑓)𝑅(𝑓).

1. Introduction

Let (𝑋,𝑑) be a metric space. For any 𝑌𝑋, denote by 𝑌, 𝜕𝑌, and 𝑌 the interior, the boundary, and the closure of 𝑌 in 𝑋, respectively. For any 𝑦𝑋 and any 𝑟>0, write 𝐵(𝑦,𝑟)={𝑥𝑋𝑑(𝑥,𝑦)<𝑟}. Let be the set of all positive integers and +={0}.

Denote by 𝐶0(𝑋) the set of all continuous maps from 𝑋 to 𝑋. For any 𝑓𝐶0(𝑋), let 𝑓0 be the identity map of 𝑋 and 𝑓𝑛=𝑓𝑓𝑛1 the composition map of 𝑓 and 𝑓𝑛1. A point 𝑥𝑋 is called a periodic point of 𝑓 with period 𝑛 if 𝑓𝑛(𝑥)=𝑥 and 𝑓𝑖(𝑥)𝑥 for 1𝑖<𝑛. The orbit of 𝑥 under 𝑓 is the set 𝑂(𝑥,𝑓){𝑓𝑛(𝑥)𝑛+}. Write 𝜔(𝑥,𝑓)=𝑖=1𝑂(𝑓𝑖(𝑥),𝑓), called the 𝜔-limit set of 𝑥 under 𝑓. In fact, 𝑦𝜔(𝑥,𝑓) if and only if there exists a sequence of positive integers 𝑛1<𝑛2<𝑛3< such that lim𝑖𝑓𝑛𝑖(𝑥)=𝑦. 𝑥 is called a recurrent point of 𝑓 if 𝑥𝜔(𝑥,𝑓). 𝑥 is called a special 𝛼-limit point of 𝑓 if there exist a sequence of positive integers {𝑛𝑖}𝑖=1 and a sequence of points {𝑦𝑖}𝑖=0 such that 𝑓𝑛𝑖(𝑦𝑖)=𝑦𝑖1 for any 𝑖 and lim𝑖𝑦𝑖=𝑥. Denote by 𝑃(𝑓), 𝑅(𝑓), and SA(𝑓) the sets of periodic points, recurrent points, and special 𝛼-limit points of 𝑓, respectively. From the definitions it is easy to see that 𝑃(𝑓)SA(𝑓) and 𝑃(𝑓)𝑅(𝑓). Let (𝑓) denote the topological entropy of 𝑓, for the definition see [1, Chapter VIII].

A metric space 𝑋 is called an arc (resp., an open arc, a circle ) if it is homeomorphic to the interval [0,1] (resp., the open interval (0,1), the unit circle 𝑆1). Let 𝐴 be an arc and [0,1]𝐴 a homeomorphism. The points (0) and (1) are called the endpoints of 𝐴, and we write End(𝐴)={(0),(1)}. A compact connected metric space 𝐺 is called a graph if there are finitely many arcs 𝐴1,,𝐴𝑛 (𝑛1) in 𝐺 such that 𝐺=𝑛𝑖=1𝐴𝑖 and 𝐴𝑖𝐴𝑗=End(𝐴𝑖)End(𝐴𝑗) for all 1𝑖<𝑗𝑛. A graph 𝑇 is called a tree if it contains no circle. A continuous map from a graph (resp., a tree, an interval) to itself is called a graph map (resp., a tree map, an interval map).

Let 𝐺 be a given graph. Take a metric 𝑑 on 𝐺 such that, for any 𝑥𝐺 and any 𝑟>0, the open ball 𝐵(𝑥,𝑟){𝑦𝐺𝑑(𝑦,𝑥)<𝑟} is always connected. For any finite set 𝑆, let |𝑆| denote the number of elements of 𝑆. For any 𝑥𝐺, write val(𝑥)=lim𝑟+0|𝜕𝐵(𝑥,𝑟)|, which is called the valence of 𝑥. 𝑥 is called a branching point (resp., an endpoint) of 𝐺 if val(𝑥)>2 (resp., val(𝑥)=1). Denote by End(𝐺) and Br(𝐺) the sets of endpoints and branching points of 𝐺, respectively. Take a finite subset 𝑉(𝐺) of 𝐺 containing End(𝐺)Br(𝐺) such that, for any connected component 𝐸 of 𝐺𝑉(𝐺), the closure 𝐸 is an arc. Such a subset 𝑉(𝐺) is called the set of vertexes of 𝐺, and the closure of every connected component of 𝐺𝑉(𝐺) is called an edge. For any edge 𝐼 of 𝐺 and any 𝑎,𝑏𝐼, we denote by [𝑎,𝑏]𝐼 (or simply [𝑎,𝑏] if there is no confusion) the smallest connected closed subset of 𝐼 containing {𝑎,𝑏}, which is called a closed interval of 𝐺. So, a closed interval is always a subset of an edge. Write (𝑎,𝑏]=[𝑏,𝑎)=[𝑎,𝑏]{𝑎} and (𝑎,𝑏)=(𝑎,𝑏]{𝑏}. Let 𝐺 be a graph and 𝐽,𝐾𝐺 closed intervals, and 𝑓𝐶0(𝐺). We write 𝑓(𝐽)𝐾 if there exists a closed subinterval 𝐿𝐽 such that 𝑓(𝐿)=𝐾.

In the study of dynamical systems, recurrent points, topological entropy, and special 𝛼-limit points play an important role. For interval maps, Hero [2] obtained the following result.

Theorem A (see [2, Corollary]). Let 𝐼 be a compact interval and 𝑓𝐶0(𝐼). Then the following are equivalent: (1)some point 𝑦 that is not recurrent is a special 𝛼-limit point;(2)some periodic point has period that is not a power of two.

It is known [1, Chapter VIII, Proposition 34] that (𝑓)>0 if and only if some periodic point of 𝑓 has period that is not a power of two for interval map 𝑓.

In [3], Llibre and Misiurewicz studied the topological entropy of a graph map and obtained the following theorem.

Theorem B (see [3, Theorems 1 and 2]). Let 𝐺 be a graph and 𝑓𝐶0(𝐺). Then (𝑓)>0 if and only if there exist 𝑛 and closed intervals 𝐿,𝐽,𝐾𝐺 with 𝐽,𝐾𝐿 and |𝐾𝐽|1 such that 𝑓𝑛(𝐽)𝐿 and 𝑓𝑛(𝐾)𝐿.

Recently, there has been a lot of work on the dynamics of graph maps (see [413]). In this paper, we will study the topological entropy and special 𝛼-limit points of graph maps. Our main result is the following theorem.

Theorem 1.1. Let 𝐺 be a graph and 𝑓𝐶0(𝐺). Then (𝑓)>0 if and only if 𝑆𝐴(𝑓)𝑅(𝑓).

2. Proof of Theorem 1.1

In this section, we will prove Theorem 1.1. To do this, we need the following lemmas.

Lemma 2.1 (see [11, Theorem 1]). Let 𝐺 be a graph and 𝑓𝐶0(𝐺). If 𝑥SA(𝑓), then there exist a sequence of positive integers 𝑛1𝑛2𝑛3 and a sequence of points {𝑦𝑖}𝑖=0 with 𝑦0=𝑥 such that 𝑓𝑛𝑖(𝑦𝑖)=𝑦𝑖1 for any 𝑖 and lim𝑖𝑦𝑖=𝑥.

Remark 2.2. The main idea of the proof of Theorem 1 in [11] is similar to the one of Main Theorem in [2].

Lemma 2.3. Let 𝐺 be a graph and 𝑓𝐶0(𝐺). Then SA(𝑓)𝑓(SA(𝑓)).

Proof. Let 𝑥SA(𝑓). Then there exist a sequence of points {𝑥𝑖}𝑖=0 and a sequence of positive integers 2𝑚1𝑚2 such that 𝑓𝑚𝑖(𝑥𝑖)=𝑥𝑖1 for every 𝑖 and lim𝑖𝑥𝑖=𝑥. Write 𝑦𝑖=𝑓𝑚𝑖1(𝑥𝑖) for 𝑖. Let 𝑦𝑘0=𝑦1,𝑦𝑘1,𝑦𝑘2,,𝑦𝑘𝑖, be a convergence subsequence of {𝑦𝑖}𝑖=1, and let lim𝑖𝑦𝑘𝑖=𝑦. Then 𝑓(𝑦)=lim𝑖𝑓𝑦𝑘𝑖=lim𝑖𝑓𝑚𝑘𝑖𝑥𝑘𝑖=lim𝑖𝑥𝑘𝑖1=𝑥.(2.1) Write 𝜇𝑖=𝑚𝑘11++𝑚1,if𝑚𝑖=1,𝑘𝑖1+𝑚𝑘𝑖2++𝑚𝑘𝑖1,if𝑖2.(2.2) Then 𝑓𝜇𝑖(𝑦𝑘𝑖)=𝑓𝜇𝑖+𝑚𝑘𝑖1(𝑥𝑘𝑖)=𝑓𝑚𝑘𝑖11(𝑥𝑘𝑖1)=𝑦𝑘𝑖1 for any 𝑖, which implies that 𝑦SA(𝑓) and SA(𝑓)𝑓(SA(𝑓)). The proof is completed.

Lemma 2.4 (see [3, Lemma 2.4]). Let 𝐺 be a graph and 𝑓𝐶0(𝐺). Suppose that 𝐽 and 𝐿=[𝑎,𝑏] are intervals of 𝐺. If there exist 𝑥(𝑎,𝑏) and 𝑦(𝑎,𝑏) such that {𝑥,𝑦}𝑓(𝐽), then 𝑓(𝐽)[𝑎,𝑥] or 𝑓(𝐽)[𝑥,𝑏].

Theorem 2.5. Let 𝐺 be a graph and 𝑓𝐶0(𝐺). Then (𝑓)>0 if and only if SA(𝑓)𝑅(𝑓).

Proof Necessity
If SA(𝑓)𝑅(𝑓), then take a point 𝑤0SA(𝑓)𝑅(𝑓). By Lemma 2.3 and 𝑓(𝑅(𝑓))=𝑅(𝑓), for every 𝑖=1,2,, there exists a point 𝑤𝑖SA(𝑓)𝑅(𝑓) such that 𝑓(𝑤𝑖)=𝑤𝑖1. Note that 𝑤0,𝑤1,𝑤2, are mutually different. Since the numbers of vertexes and edges of 𝐺 are finite, there exists an edge 𝐼 of 𝐺 such that 𝐼{𝑤0,𝑤1,𝑤2,} is an infinite set. We can choose integers 1<𝑖1<𝑖2< such that {𝑤𝑖𝑘𝑘}𝐼 and 𝑤𝑖𝑘(𝑤𝑖1,𝑤𝑖𝑘+1) for every 𝑘2. Take points {𝑦,𝑥,𝑧}𝐼(SA(𝑓)𝑅(𝑓)) with 𝑥(𝑦,𝑧) such that 𝑓𝑚(𝑦)=𝑥 and 𝑓𝑛(𝑥)=𝑧 for some 𝑚,𝑛. Without loss of generality we may assume that 𝐼=[0,1] and 0<𝑦<𝑥<𝑧<1. Since 𝑦SA(𝑓)𝑅(𝑓), we can take points {𝑦𝑖𝑖}(0,1) and positive integers 𝑚+𝑛<𝑚1<𝑚2<𝑚3< satisfying the following conditions:(1)the sequence (𝑦1,𝑦2,𝑦3,) is strictly monotonic with 𝑓𝑚𝑖(𝑦𝑖)=𝑦𝑖1 for any 𝑖 and 𝑦0=𝑦 (see Lemma 2.1) and lim𝑖𝑦𝑖=𝑦;(2)𝑚𝑖>𝑚1+𝑚2++𝑚𝑖1 for any 𝑖2.
Let 𝑥𝑖=𝑓𝑚(𝑦𝑖) and 𝑧𝑖=𝑓𝑛(𝑥𝑖) for any 𝑖+. Then lim𝑖𝑥𝑖=𝑥 and lim𝑖𝑧𝑖=𝑧. Noting that 𝑥,𝑧SA(𝑓)𝑅(𝑓), we can assume that {𝑥𝑖,𝑧𝑖𝑖}(0,1), and there exists 𝜀>0 such that the following conditions hold:
(3)𝑓𝑖(𝑥)[𝑥𝜀,𝑥+𝜀] for any 𝑖;(4)the sequences (𝑥1,𝑥2,𝑥3,) and (𝑧1,𝑧2,𝑧3,) are strictly monotonic, and {𝑥𝑖𝑖}[𝑥𝜀,𝑥+𝜀](𝑦,𝑧).
In the following we may consider only the case that (𝑥1,𝑥2,𝑥3,) is strictly decreasing since the other case that (𝑥1,𝑥2,𝑥3,) is strictly increasing is similar.
Write 𝜇𝑖=𝑚𝑖+𝑚𝑖1++𝑚1 for any 𝑖. Put 𝐼𝑖=[𝑥𝑖,𝑥𝑖1] and 𝐴𝑖=𝑓𝜇𝑖1(𝐼𝑖) for any 𝑖2. Then 𝐴𝑖 is a connected set, and𝑓𝜇𝑖1𝑥𝑖1,𝑓𝜇𝑖1𝑥𝑖=𝑥,𝑓𝜇𝑖1𝑥𝑖𝐴𝑖.(2.3) Noting that 𝑓𝑚𝑖(𝑓𝜇𝑖1(𝑥𝑖))=𝑓𝜇𝑖(𝑥𝑖)=𝑥, we have 𝑥𝑓𝑚𝑖(𝐴𝑖)𝐴𝑖. Write 𝑆𝑖=𝑗=0𝑓𝑗𝑚𝑖(𝐴𝑖). Then 𝑆𝑖 is a connected set containing 𝑥 and 𝑓𝑚𝑖(𝑆𝑖)𝑆𝑖 for every 𝑖2.
Since 𝑓𝑚𝑖(𝑥𝑖1)=𝑓𝑚𝑖𝜇𝑖1(𝑥) and 𝑓𝑚𝑖(𝑥𝑖)=𝑥𝑖1 for any 𝑖2, by Lemma 2.4 it follows that 𝑓𝑚𝑖(𝐼𝑖)[𝑥𝜀,𝑥𝑖1] or 𝑓𝑚𝑖(𝐼𝑖)[𝑥𝑖1,𝑥+𝜀]. There are two cases to consider.
Case 1. There exist 2𝛼<𝛽<𝜆 such that 𝑓𝑚𝑖(𝐼𝑖)[𝑥𝜀,𝑥𝑖1] for every 𝑖{𝛼,𝛽,𝜆}.
Subcase 1.1. There exists 𝜆𝜏 such that 𝑆𝜏(0,1). Then 𝑆𝜏{𝑦𝛼,𝑧𝛼+1}, and there exist 𝑟𝜇𝜏1 and 𝑢𝐼𝜏 such that 𝑓𝑟(𝑢){𝑦𝛼,𝑧𝛼+1}, from which and 𝑚𝛼+1>𝑚+𝑛 it follows 𝑓𝑚+𝑟(𝑢)=𝑓𝑚𝑦𝛼=𝑥𝛼or𝑓𝑚𝛼+1𝑛+𝑟(𝑢)=𝑓𝑚𝛼+1𝑛𝑧𝛼+1=𝑥𝛼.(2.4) Noting 𝑓𝑚+𝑟(𝑥𝜏1)=𝑓𝑚+𝑟𝜇𝜏1(𝑥) and 𝑓𝑚𝛼+1𝑛+𝑟(𝑥𝜏1)=𝑓𝑚𝛼+1𝑛+𝑟𝜇𝜏1(𝑥), we have 𝑓𝑚+𝑟𝜇𝜏1(𝑥),𝑓𝑚𝛼+1𝑛+𝑟𝜇𝜏1[](𝑥)𝑥𝜀,𝑥+𝜀=.(2.5) There exists 𝑠{𝑚+𝑟,𝑚𝛼+1𝑛+𝑟} such that 𝑓𝑠(𝐼𝜏)𝐼𝛽𝐼𝜆 or 𝑓𝑠(𝐼𝜏)𝐼𝛼, which implies 𝑓𝑠+𝑚𝜆𝐼𝜆𝑓𝑠𝐼𝜏𝐼𝛽𝐼𝜆or𝑓𝑠+𝑚𝛼+𝑚𝜆𝐼𝜆𝑓𝑠+𝑚𝛼𝐼𝜏𝑓𝑚𝛼𝐼𝛼𝐼𝛽𝐼𝜆.(2.6) On the other hand, 𝑓𝑚𝛽(𝐼𝛽)𝐼𝛽𝐼𝜆. Thus we can obtain 𝑓𝑙(𝐼𝜆)𝐼𝛽𝐼𝜆 and 𝑓𝑙(𝐼𝛽)𝐼𝛽𝐼𝜆 for some 𝑙{(𝑠+𝑚𝜆)𝑚𝛽,(𝑠+𝑚𝛼+𝑚𝜆)𝑚𝛽}. By Theorem B it follows that (𝑓)>0.
Subcase 1.2. 𝑆𝑖(0,1) for all 𝑖𝜆, and there exists 𝜏𝜆 such that 𝑥<sup𝑆𝜏. Then we can take 𝑗𝜏 such that [𝑥,𝑥𝑗]𝑆𝜏. Thus there exist 𝑟𝜇𝜏1 and 𝑢𝐼𝜏 such that 𝑓𝑟(𝑢)=𝑥𝑗, which implies 𝑓𝑟+𝑚𝑗++𝑚𝛼+1(𝑢)=𝑥𝛼. Write 𝑠=𝑟+𝑚𝑗++𝑚𝛼+1. Then 𝑓𝑠(𝐼𝜏)𝐼𝛽𝐼𝜆 or 𝑓𝑠(𝐼𝜏)𝐼𝛼 since 𝑓𝑠(𝑥𝜏1)=𝑓𝑠𝜇𝜏1(𝑥)[𝑥𝜀,𝑥+𝜀], which implies 𝑓𝑠+𝑚𝜆𝐼𝜆𝑓𝑠𝐼𝜏𝐼𝛽𝐼𝜆or𝑓𝑠+𝑚𝛼+𝑚𝜆𝐼𝜆𝑓𝑠+𝑚𝛼𝐼𝜏𝑓𝑚𝛼𝐼𝛼𝐼𝛽𝐼𝜆.(2.7) On the other hand, 𝑓𝑚𝛽(𝐼𝛽)𝐼𝛽𝐼𝜆. Thus we can obtain 𝑓𝑙(𝐼𝜆)𝐼𝛽𝐼𝜆 and 𝑓𝑙(𝐼𝛽)𝐼𝛽𝐼𝜆 for some 𝑙{(𝑠+𝑚𝜆)𝑚𝛽,(𝑠+𝑚𝛼+𝑚𝜆)𝑚𝛽}. By Theorem B it follows that (𝑓)>0.
Subcase 1.3. One has𝑆𝑖(0,1) and 𝑥=sup𝑆𝑖 for all 𝑖𝜆.
If 𝑓𝑚𝑟(𝑥)<𝑓2𝑚𝑟(𝑥)<𝑥 for some 𝑟𝜆, then there exist 𝑗𝑟+2 and 𝑢𝐼𝑟 such that 𝑓𝜇𝑟(𝑢)=𝑓2𝑚𝑟(𝑥𝑗) since lim𝑖𝑓2𝑚𝑟(𝑥𝑖)=𝑓2𝑚𝑟(𝑥) and {𝑓𝑚𝑟(𝑥),𝑥}𝑓𝜇𝑟(𝐼𝑟), which implies 𝑓𝜇𝑟+𝑚𝑗+𝑚𝑗1++𝑚𝛼+12𝑚𝑟(𝑢)=𝑥𝛼. Using arguments similar to ones developed in the proof of Subcase 1.2, we can obtain 𝑓𝑙(𝐼𝜆)𝐼𝛽𝐼𝜆 and 𝑓𝑙(𝐼𝛽)𝐼𝛽𝐼𝜆 for some 𝑙. By Theorem B it follows that (𝑓)>0. Now we assume 𝑓2𝑚𝑟(𝑥)𝑓𝑚𝑟(𝑥)<𝑥 for all 𝑟𝜆. Note 𝑓𝜇𝑟1(𝑥𝑟)𝑂(𝑓𝑚𝑟,𝑥) since 𝑥𝑅(𝑓).
If 𝑓2𝑚𝑟(𝑥)𝑓𝑚𝑟(𝑥)<𝑓𝜇𝑟1(𝑥𝑟)<𝑥 for some 𝑟𝜆, then 𝑓𝑚𝑟([𝑓𝑚𝑟(𝑥),𝑓𝜇𝑟1(𝑥𝑟)])[𝑓𝑚𝑟(𝑥),𝑥] and 𝑓𝑚𝑟([𝑓𝜇𝑟1(𝑥𝑟),𝑥])[𝑓𝑚𝑟(𝑥),𝑥]. By Theorem B it follows that (𝑓)>0.
If 𝑓𝜇𝑟1(𝑥𝑟)<𝑓𝑚𝑟(𝑥) for some 𝑟𝜆, then there exist 𝑗𝑟+2 and 𝑢𝐼𝑟 such that 𝑓𝜇𝑟1(𝑢)=𝑓𝑚𝑟(𝑥𝑗) since lim𝑖𝑓𝑚𝑟(𝑥𝑖)=𝑓𝑚𝑟(𝑥) and {𝑓𝜇𝑟1(𝑥𝑟),𝑥}𝑓𝜇𝑟1(𝐼𝑟), which implies 𝑓𝜇𝑟1+𝑚𝑗+𝑚𝑗1++𝑚𝛼+1𝑚𝑟(𝑢)=𝑥𝛼. Using arguments similar to ones developed in the proof of Subcase 1.2, we can obtain 𝑓𝑙(𝐼𝜆)𝐼𝛽𝐼𝜆 and 𝑓𝑙(𝐼𝛽)𝐼𝛽𝐼𝜆 for some 𝑙. By Theorem B it follows that (𝑓)>0.
Case 2. There exists 𝜅2 such that 𝑓𝑚𝑖(𝐼𝑖)[𝑥𝑖1,𝑥+𝜀] for all 𝑖𝜅.
Subcase 2.1. There exist 𝜅𝛼<𝛽 such that 𝑆𝑖(0,1) for every 𝑖{𝛼,𝛽}. Then 𝑆𝛽{𝑦𝛽,𝑧𝛽+1} and 𝑆𝛼{𝑦𝛽,𝑧𝛽+1}. Thus there exist 𝑟𝜇𝛽1 and 𝑢𝐼𝛽 such that 𝑓𝑟(𝑢){𝑦𝛽,𝑧𝛽+1}, from which it follows that 𝑓𝑚+𝑟(𝑢)=𝑥𝛽 or 𝑓𝑚𝛽+1𝑛+𝑟(𝑢)=𝑥𝛽. Since 𝑓𝑚+𝑟(𝑥𝛽1)=𝑓𝑚+𝑟𝜇𝛽1(𝑥), 𝑓𝑚𝛽+1𝑛+𝑟(𝑥𝛽1)=𝑓𝑚𝛽+1𝑛+𝑟𝜇𝛽1(𝑥), and 𝑓𝑚+𝑟𝜇𝛽1(𝑥),𝑓𝑚𝛽+1𝑛+𝑟𝜇𝛽1[](𝑥)𝑥𝜀,𝑥+𝜀=,(2.8) there exists 𝑠{𝑚+𝑟,𝑚𝛽+1𝑛+𝑟} such that 𝑓𝑠(𝐼𝛽)𝐼𝛽𝐼𝛼 or 𝑓𝑠(𝐼𝛽)𝐼𝛽+1, which implies 𝑓𝑠(𝐼𝛽)𝐼𝛽𝐼𝛼 or 𝑓𝑠+𝑚𝛽+1(𝐼𝛽)𝑓𝑚𝛽+1(𝐼𝛽+1)𝐼𝛽𝐼𝛼. In similar fashion, we can show 𝑓𝑡(𝐼𝛼)𝐼𝛽𝐼𝛼 for some 𝑡. Thus we get 𝑓𝑙(𝐼𝛽)𝐼𝛽𝐼𝛼 and 𝑓𝑙(𝐼𝛼)𝐼𝛽𝐼𝛼 for some 𝑙{𝑠𝑡,(𝑠+𝑚𝛽+1)𝑡}. It follows from Theorem B that (𝑓)>0.
Subcase 2.2. There exists 𝜗𝜅 such that 𝑆𝑖(0,1) for all 𝑖𝜗 and there exists 𝜏𝜆𝜗 such that 𝑥<sup𝑆𝜏 and 𝑥<sup𝑆𝜆. Take 𝑗𝜏+2 such that 𝑆𝑖[𝑥,𝑥𝑗] for 𝑖{𝜆,𝜏}. Then there exist 𝑟1𝜇𝜏1,𝑟2𝜇𝜆1, and 𝑢𝐼𝜏,𝑣𝐼𝜆 such that 𝑓𝑟1(𝑢)=𝑓𝑟2(𝑣)=𝑥𝑗. Using arguments similar to ones developed in the proof of Subcase 2.1, we can obtain 𝑓𝑙(𝐼𝜆)𝐼𝜏𝐼𝜆 and 𝑓𝑙(𝐼𝜏)𝐼𝜏𝐼𝜆 for some 𝑙. By Theorem B it follows that (𝑓)>0.
Subcase 2.3. There exists 𝜗𝜅 such that 𝑆𝑖(0,1) and 𝑥=sup𝑆𝑖 for all 𝑖𝜗.
If there exist 𝜏>𝜆𝜗 such that 𝑓𝑚𝑖(𝑥)<𝑓2𝑚𝑖(𝑥)<𝑥 for 𝑖{𝜏,𝜆}, then there exist 𝑗𝜏+2, 𝑢𝐼𝜏, and 𝑣𝐼𝜆 such that 𝑓𝜇𝜏(𝑢)=𝑓2𝑚𝜏(𝑥𝑗) and 𝑓𝜇𝜆(𝑣)=𝑓2𝑚𝜆(𝑥𝑗), which implies 𝑓𝜇𝜏+𝑚𝑗+𝑚𝑗1++𝑚𝜏+12𝑚𝜏(𝑢)=𝑥𝜏 and 𝑓𝜇𝜆+𝑚𝑗+𝑚𝑗1++𝑚𝜏+12𝑚𝜆(𝑣)=𝑥𝜏. Using arguments similar to ones developed in the proof of Subcase 2.1, we can obtain 𝑓𝑙(𝐼𝜆)𝐼𝜏𝐼𝜆 and 𝑓𝑙(𝐼𝜏)𝐼𝜏𝐼𝜆 for some 𝑙. By Theorem B it follows that (𝑓)>0. Now we assume that there exists 𝜃𝜗 such that 𝑓2𝑚𝑖(𝑥)𝑓𝑚𝑖(𝑥)<𝑥 for all 𝑖𝜃.
If 𝑓𝜇𝑖1(𝑥𝑖)<𝑓𝑚𝑖(𝑥)<𝑥 for all 𝑖𝜃, then using arguments similar to ones developed in the above proof, we can obtain (𝑓)>0.
If 𝑓2𝑚𝑟(𝑥)𝑓𝑚𝑟(𝑥)<𝑓𝜇𝑟1(𝑥𝑟)<𝑥 for some 𝑟𝜃, then 𝑓𝑚𝑟([𝑓𝑚𝑟(𝑥),𝑓𝜇𝑟1(𝑥𝑟)])[𝑓𝑚𝑟(𝑥),𝑥] and 𝑓𝑚𝑟([𝑓𝜇𝑟1(𝑥𝑟),𝑥])[𝑓𝑚𝑟(𝑥),𝑥]. By Theorem B it follows (𝑓)>0.

Sufficiency
If (𝑓)>0, then it follows from Theorem B that there exist 𝑛 and closed intervals 𝐿,𝐽,𝐾𝐺 with 𝐽,𝐾𝐿 and |𝐾𝐽|1 such that 𝑓𝑛(𝐽)=𝐿 and 𝑓𝑛(𝐾)=𝐿. Without loss of generality we may assume that 𝐿=[0,1] and 𝐽=[𝑎,𝑏] and 𝐾=[𝑐,𝑑] with 0𝑎<𝑏𝑐<𝑑1 such that 𝑓𝑛([𝑎,𝑏])=[0,1] and 𝑓𝑛([𝑐,𝑑])=[0,1]. By [1, Chapter II, Lemma 2] we can choose 𝑢,𝑣,𝑤[0,1] with 𝑢<𝑣<𝑤 such that one of the following statements holds:(i)𝑓𝑛(𝑢)=𝑓𝑛(𝑤)=𝑢, 𝑓𝑛(𝑣)=𝑤, 𝑓𝑛(𝑥)>𝑢 for 𝑢<𝑥<𝑤 and 𝑥<𝑓𝑛(𝑥)<𝑤 for 𝑢<𝑥<𝑣.(ii)𝑓𝑛(𝑢)=𝑓𝑛(𝑤)=𝑤, 𝑓𝑛(𝑣)=𝑢, 𝑓𝑛(𝑥)<𝑤 for 𝑢<𝑥<𝑤 and 𝑢<𝑓𝑛(𝑥)<𝑥 for 𝑣<𝑥<𝑤.
We may consider only case (i) since case (ii) is similar. We claim that, for any 𝑥(𝑣,𝑤) and any 0<𝜀<𝑤𝑥, there exist 𝑦[𝑤𝜀,𝑤) and 𝑠 such that 𝑓𝑠𝑛(𝑦)=𝑥. In fact, we can choose 𝑢<<𝑥𝑖<𝑥𝑖1<<𝑥1𝑣<𝑥0=𝑥 such that lim𝑖𝑥𝑖=𝑢 and 𝑓𝑛(𝑥𝑖)=𝑥𝑖1 for any 𝑖. Thus there exists some 𝑥𝑁𝑓𝑛([𝑤𝜀,𝑤)). That is, we can choose 𝑦[𝑤𝜀,𝑤) satisfying 𝑓𝑛(𝑦)=𝑥𝑁, which implies 𝑓(𝑁+1)𝑛(𝑦)=𝑥. The claim is proven.
By the above claim we can choose a sequence of positive integers {𝑠𝑖}𝑖=1 and a sequence of points 𝑣<𝑦0<𝑦1<𝑦2<<𝑤 such that 𝑓𝑛𝑠i(𝑦𝑖)=𝑦𝑖1 for any 𝑖 and lim𝑖𝑦𝑖=𝑤. Note that 𝑓𝑛(𝑤)=𝑓𝑛(𝑢)=𝑢; then 𝑤SA(𝑓𝑛)𝑅(𝑓𝑛)SA(𝑓)𝑅(𝑓). The proof is completed.

Acknowledgments

Project Supported by NSF of China (10861002) and NSF of Guangxi (2010GXNSFA013106, 2011GXNSFA018135) and SF of Education Department of Guangxi (200911MS212).

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