A. N. Sharkovsky

On Attracting and Attracted Sets

(Presented by Academician P. S. Aleksandrov, 21 IX 1964)

Let \(T\) be a continuous single-valued mapping of a compact space \(E\) into itself. Every point \(x \in E\) generates the iterative sequence \(\{T^j x\}_{j=0}^{\infty}\). If the points \(x, Tx, \ldots, T^{k-1}x\) are pairwise distinct and \(T^{k-1}x = x\), then the points \(x, Tx, \ldots, T^{k-1}x\) form a cycle of order \(k\).

A point \(y\) is called an \(\omega\)-limit point of the sequence \(\{T^j x\}\) if, for every neighborhood \(U\) of the point \(y\) and every \(n > 0\), there exists a number \(m \ge n\) such that \(T^m x \in U\). We shall denote the set of \(\omega\)-limit points of the sequence \(\{T^j x\}\) by \(\Omega_x\).

1. The set \(\Omega = \Omega_x\) is closed and \(T\Omega = \Omega\). The following two theorems characterize the mapping \(T\) on the set \(\Omega\).

Theorem 1. If \(U\) is a set open in \(\Omega\), and \(U \ne \Omega\), then the closure of the set \(TU\) is not contained in \(U\).

From Theorem 1 it follows immediately that

Corollary 1. If \(\Omega' \subset \Omega\) is such that \(T\Omega' = \Omega'\), then \(\Omega'\) cannot be both closed and open in \(\Omega\).

Corollary 2. If \(\Omega\) is finite, the points of the set \(\Omega\) form a cycle.

Corollary 3. If \(\Omega\) is infinite, every point of a cycle belonging to \(\Omega\) is a limit point for the points of the set \(\Omega\).

Theorem 2. If the set \(\Omega\) is infinite and \(T^j x \in \Omega\) for \(j \ge j_0\), then for any set \(U\) open in \(\Omega\), the derived set of the set

\[ \bigcup_{j=0}^{\infty} T^j U \]

is \(\Omega\).

If the set \(\Omega\) has interior points in \(E\), then a number \(j_0\) such that \(T^j x \in \Omega\) for \(j \ge j_0\) always exists.

Theorems 1 and 2 completely characterize the mapping on the set \(\Omega\). By this is meant the following.

A. If a continuous mapping \(T\) of a compact space \(E\) into itself is given such that on the closed set \(\Omega\), \(T\Omega = \Omega\), and the assertion of Theorem 2 holds, then there exists a point \(x \in \Omega\) such that \(\Omega_x = \Omega\).

B. If a closed set \(\Omega\) has no interior points in \(E\), contains no isolated points of \(E\), and a continuous mapping \(T\) is given on it such that \(T\Omega = \Omega\) and Theorem 1 holds, then the mapping \(T\) can be extended to a closed set \(E'\), \(\Omega \subset E' \subseteq E\), so that the mapping \(T\) on \(E'\) is continuous and there exists a point \(x \in E'\) for which \(\Omega_x = \Omega\).

The question of the structure of the set of \(\omega\)-limit points of an iterative sequence is reduced to the following: what is the structure of a closed set \(\Omega \subseteq E\), if on the set \(\Omega\) one can define a continuous mapping \(T\) such that \(T\Omega = \Omega\) and Theorem 1 or 2 holds.

One may note the following result: if the compact space \(E\) is locally connected and the set \(\Omega\) has interior points in \(E\), then

\[ \Omega = \bigcup_{j=1}^{k} \Omega^{(j)}, \quad 1 \le k < \infty, \]

where \(\Omega^{(1)}, \ldots, \Omega^{(k)}\) are connected closed sets having no common points, and \(T\Omega^{(j)} = \Omega^{(j+1)}\), \(j = 1, 2, \ldots, k - 1\), \(T\Omega^{(k)} = \Omega^{(1)}\).

1. We shall say that a point \(x\in E\) is attracted by a set \(\Omega\) if \(\Omega\) is the set of \(\omega\)-limit points of the sequence
   \[ \{T^{j}x\}_{j=0}^{\infty}. \]
   The set consisting of the points of the compactum that are attracted to a given set \(\Omega\) will be denoted by \(P(\Omega)\). In what follows we shall agree to denote by \(\Omega\) (with indices or without them) only those sets for which \(P(\Omega)\) is nonempty.

Theorem 3.
\[ \bigcup_{\Omega'\supseteq \Omega} P(\Omega') \]
is a set of type \(G_\delta\).

Let \(\Sigma\) be a system of open sets \(\sigma_i,\ i=1,2,\ldots,\) such that: 1) \(\sigma_i\cap\Omega\ne 0,\ i=1,2,\ldots;\) 2) for every neighborhood \(U\) of a point \(x\in\Omega\) there is a \(\sigma\in\Sigma\) contained in \(U\). For each \(\sigma_i\in\Sigma\) construct the open set \(S(\sigma_i)\), consisting of all preimages of the set \(\sigma_i\): \(x\in S(\sigma_i)\) if and only if there exists \(k\geq 0\) such that
\[ T^{k}x\in\sigma_i. \]
Then
\[ \bigcap_{i=1}^{\infty} S(\sigma_i)=\bigcup_{\Omega'\supseteq\Omega}P(\Omega'). \]
Indeed, if \(x\in P(\Omega')\), where \(\Omega'\supseteq\Omega\), then \(x\in S(\sigma_i)\), \(i=1,2,\ldots\). If \(x\in P(\Omega'')\) and \(\Omega''\) does not contain \(\Omega\), i.e. there exists a point \(y\in\Omega\) not belonging to \(\Omega''\), then there is a \(\delta_i\ni y\) containing not a single point of the sequence \(\{T^{j}x\}_{j=0}^{\infty}\). Consequently,
\[ x\notin S(\sigma_i) \]
and
\[ x\notin \bigcup_{\Omega'\supseteq\Omega} P(\Omega'). \]

Theorem 4.
\[ \bigcup_{\Omega'\subseteq\Omega}P(\Omega') \]
is a set of type \(F_{\sigma\delta}\).

Indeed, if \(F\) is an arbitrary closed set, then the set \(p(F)\), consisting of the points \(x\in E\) for which \(T^{j}x\in F\) for \(j\geq j_0\) (\(j_0\) depends on \(x\)), is a set of type \(F_\sigma\). Consider a sequence of open sets
\[ U_1\supset U_2\supset U_3\supset\ldots \]
such that
\[ \bigcap_{i=1}^{\infty}U_i=\Omega, \]
and let \(F_i\) be the closures of the sets \(U_i\). Construct the sets \(p(F_i)\), \(i=1,2,\ldots\).
\[ \bigcap_{i=1}^{\infty}p(F_i) \]
is the set
\[ \bigcup_{\Omega'\subseteq\Omega}P(\Omega'). \]
Indeed, if \(x\in P(\Omega')\), \(\Omega'\subseteq\Omega\), then \(x\in p(F_i)\), \(i=1,2,\ldots\). If \(x\in P(\Omega'')\) and there exists a point \(y\in\Omega''\) not belonging to \(\Omega\), then there is a number \(i_0\) such that \(y\notin F_{i_0}\), and then \(x\notin p(F_i)\), \(i\geq i_0\).

Corollary. \(P(\Omega)\) is a set of type \(F_{\sigma\delta}\).

If the set \(\Omega\cap P(\Omega)\) is nonempty, then \(P(\Omega)\) on the set \(\Omega\), as follows from Theorem 3, is a set of type \(G_\delta\) of the second category.

The question arises whether there exist mappings for which the sets \(P(\Omega)\) are sets of type \(G_\delta\) or \(F_\sigma\) and are not sets of a simpler type. An affirmative answer to this question is given by the theorems formulated below.

1. Consider the case when \(E\) is a segment of the real line.

Theorem 5. If \(\Omega\) is infinite, then \(P(\Omega)\) is a set of class \(\geq 1\) in the Baire—Wali-Poussin classification.

Corollary. If \(\Omega\) is infinite and there is no set \(\Omega'\supset \Omega\), then \(P(\Omega)\) is \(G_\delta\) and is not \(F_\sigma\).

Theorem 5 follows from Lemma 1.

Lemma 1. If the hypotheses of Theorem 5 are fulfilled, then: 1) in every neighborhood of each point \(x\in P(\Omega)\) there are points \(x'<x,\ x'\notin P(\Omega)\) (if \(x\) is not the left endpoint of \(E\)) and \(x''>x,\ x''\notin P(\Omega)\) (if \(x\) is not the right endpoint of \(E\)); 2) if an interval \((a,b)\subset P(\Omega)\), then also the points \(a,b\in P(\Omega)\).

Theorem 6. If \(\Omega\) contains a cycle and there exists \(\Omega'\supset \Omega\), then
\[ \bigcup_{\Omega'\supseteq\Omega} P(\Omega') \]
is a set of the second class.

The proof of Theorem 6 is based on Lemmas 1 and 2.

Lemma 2. If \((a,b)\subset \displaystyle\bigcup_{\Omega'\supseteq\Omega} P(\Omega')\) and \(\Omega\) contains a cycle, then there exists an \(\Omega'\) such that \((a,b)\subset P(\Omega')\).

Lemma 2 is apparently also true if \(\Omega\) does not contain cycles.

Theorem 7. If \(\Omega\) contains a cycle and 1) there exists \(\Omega'\supset \Omega\); 2) in every neighborhood \(U\) of the set \(\Omega\) there is a point \(x\in P(\Omega)\), \(x\notin\Omega\), \(T^j x\in U\), \(j=0,1,2,\ldots\), then the set \(P(\Omega)\) is a set of the third class in the Baire—Vallée-Poussin classification.

Thus, under the conditions of the theorem, the set \(P(\Omega)\) is \(F_{\sigma\delta}\) and is not \(G_{\delta\sigma}\).

For example, for the mapping \(Tx=x-x\sin \frac{1}{x}\) of the interval \([0,1]\), the set of points \(x\) for which \(T^j x\to 0\) as \(j\to\infty\) is precisely \(F_{\sigma\delta}\).

If the set \(\Omega\) is infinite and contains at least one isolated point (and hence also a countable set of isolated points), then condition 2) of the theorem is satisfied.

The outline of the proof of Theorem 7 is as follows: 1) in the set
\[ \bigcup_{\Omega'\supseteq\Omega} P(\Omega') \]
one chooses a subset \(J\) homeomorphic to the set of irrational numbers \((^1)\); 2) it is shown that the set \(P(\Omega)\cap J\) can be obtained in the same way and from the same elements as a Baire set of the third class \((^2)\).

As a consequence of Theorems 6 and 7 we obtain Theorem 8.

Theorem 8. If the conditions of Theorem 7 are fulfilled,
\[ \bigcup_{\Omega'\supseteq\Omega} P(\Omega') \]
is a set of the third class.

Received
8 IX 1964

CITED LITERATURE

\(^{1}\) P. S. Aleksandrov, P. S. Uryson, Math. Ann., 98 (1927). \(^{2}\) N. N. Luzin, Lectures on Analytic Sets, Collected Works, 2, Publishing House of the Academy of Sciences of the USSR, 1958, Ch. II.