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).