MATHEMATICS

A. L. GENIS

METRIC PROPERTIES OF ENDOMORPHISMS OF THE \(n\)-DIMENSIONAL TORUS

(Presented by Academician A. N. Kolmogorov, 2 II 1961)

It is known that any group endomorphism of the \(n\)-dimensional torus can be regarded as a metric endomorphism of the torus \((^3)\). In the present paper some properties of metric endomorphisms of the \(n\)-dimensional torus are established.

Let \(E^n\) be \(n\)-dimensional Euclidean space in which an orthogonal normalized basis has been chosen. It is known that \(E^n\) is an Abelian group under addition, and the set of points of \(E^n\) with integral coordinates is a subgroup, which we shall denote by \(\Delta^n\). The factor group \(E^n / \Delta^n\) is a certain torus \(T^n\). A group endomorphism \(S^n\) of the torus \(T^n\) is generated by a certain linear transformation \(\widetilde S_n\) of the space \(E^n\), which is written in the basis chosen by us as an integral matrix. The following theorem reduces the study of an arbitrary endomorphism to the study, in a known sense, of “elementary” endomorphisms.

Theorem 1. Let
\[ p(\lambda)=p_1^{\,n_1}(\lambda)\cdot \ldots \cdot p_m^{\,n_m}(\lambda) \]
be the decomposition of the characteristic polynomial of the matrix \(\widetilde S_n\) into factors irreducible over the field of rational numbers. Then the torus \(T^n\) can be decomposed into a direct product of tori \(T_i\), where the dimension of the \(i\)-th torus is \(n_i k_i\), where \(k_i\) is the degree of the polynomial \(p_i(\lambda)\), \(i=1,\ldots,m\). Each torus \(T_i\) is invariant with respect to the endomorphism \(S_n\), and if the endomorphism acting on the torus \(T_i\) is denoted by \(S_i\), then the endomorphism \(S_n\) admits the direct decomposition
\[ S_n=S_1\times S_2\times \ldots \times S_m . \]

Proof. Let \(\lambda_{11},\ldots,\lambda_{1k_1}\) be the roots of the polynomial \(p_1(\lambda)\). From the irreducibility of the polynomial it can be inferred that they are either all real, or all complex (with nonzero imaginary parts).

Consider the case when all roots \(\lambda_{1i}\) are real. Then the linear transformation \(\widetilde S_n\) has an eigenvector \(\widetilde x_{11}^{(1)}\)
\[ \widetilde S_n \widetilde x_{11}^{(1)}=\lambda_{11}\widetilde x_{11}^{(1)} . \]
The set of vectors \((\{t\widetilde x_{11}^{(1)}\},-\infty<t<\infty)\) forms a subgroup in \(E^n\). The closure of the image of this group in \(T^n\) is a closed connected subgroup of the torus \(T^n\), i.e. a certain torus of dimension \(k\). By a well-known theorem of Kronecker \((^7)\), the number \(k\) is equal to the number of rationally independent coordinates of the vector \(\widetilde x_{11}^{(1)}\). These coordinates are certain polynomials in \(\lambda_{11}\) with integral coefficients; from the irreducibility of the polynomial \(p_1(\lambda)\) it follows that there exist no more than \(k_1\) such rationally independent polynomials, i.e.
\[ k \leq k_1 . \tag{1} \]

The constructed torus \(T^k\) is invariant with respect to \(S_n\). Consequently, its complete inverse image \(E^k\) in the space \(E^n\) is invariant with respect to \(\widetilde S_n\). But \(E^k\)

there is a certain \(k\)-dimensional linear subspace, defined by \(n-k\) linear equations with integer coefficients. In it one can choose a basis so that the transformation \(\widetilde S_n\) in \(E^k\) is given by an integral matrix, and the vector \(\widetilde x_{11}^{(1)}\) will belong to \(E^k\). Therefore \(\lambda_{11}^{(1)}\) will be an eigenvalue for the transformation \(\widetilde S_n\) on \(E^k\), and hence will satisfy the characteristic polynomial of the transformation \(\widetilde S_n\) on \(E^k\). Since the degree of this polynomial is equal to \(k\), we obtain

\[ k\geq k_1. \tag{2} \]

From comparison of (1) and (2) it follows that \(k=k_1\). Therefore the characteristic polynomial of \(\widetilde S_n\) on \(E^k\) coincides with \(p_1(\lambda)\). The root \(\lambda_{11}\) has multiplicity \(n_1\), and then either there exists another eigenvector \(\widetilde x_{11}^{(2)}\) with the same eigenvalue \(\lambda_{11}\), in which case the preceding construction is applicable, or there exists a vector \(\widetilde x_{11}^{(2)}\) satisfying the equation

\[ \widetilde S_n \widetilde x_{11}^{(2)}=\lambda_{11}\widetilde x_{11}^{(2)}+\widetilde x_{11}^{(1)}. \]

We construct, from the vector \(\widetilde x_{11}^{(2)}\), the torus \(T_1^{k_1}\) and the subspace \(E_1^{k_1}\) analogously to the torus \(T^{k_1}\) and the subspace \(E^{k_1}\). It is easy to see that \(T^{2k_1}=T^{k_1}\times T_1^{k_1}\) is a torus invariant with respect to the endomorphism \(S_n\), and the characteristic polynomial of the transformation \(\widetilde S_n\) on \(E^{k_1}\times E_1^{k_1}\) is \(p_1^2(\lambda)\). Carrying out such a construction with all the vectors \(\widetilde x_{11}^{(i)}\), we obtain a torus \(T^{n_1k_1}\) invariant with respect to \(S_n\) and a Euclidean space \(E^{n_1k_1}\) invariant with respect to \(\widetilde S_n\). The characteristic polynomial of \(\widetilde S_n\) on \(E^{n_1k_1}\) is \(p_1^{n_1}(\lambda)\). Constructing tori for all polynomials \(p_i^{n_i}(\lambda)\), we obtain the required decomposition. The case of imaginary roots differs in no way from the one considered. The theorem is proved.

Let us recall the definitions (see \((^{1,4,5})\)).

Definition 1. A metric automorphism \(T\) of a Lebesgue space \(M\) is called a Kolmogorov automorphism if there exists a measurable partition \(\xi\) with the following properties:

1) \(\xi<T\xi\); 2) \(\displaystyle \prod_{n>0}^{\infty} T^n\xi=\varepsilon \bmod 0\), \(\varepsilon\) is the partition into individual points;

3) \(\displaystyle \bigcup_{n=0}^{-\infty} T^n\xi=\nu \bmod 0\), \(\nu\) is the trivial partition, whose only element is all of \(M\).

Definition 2. A metric endomorphism \(T\) of a Lebesgue space \(M\) is called an exact endomorphism if \(\displaystyle \bigcap_{n=0}^{\infty} T^{-n}\varepsilon=\nu\).

Theorem 2. If the characteristic polynomial \(p(\lambda)\) of an endomorphism \(S_n\) of the torus \(T^n\) is representable in the form \(p(\lambda)=\varphi^s(\lambda)\), where \(s\) is an integer and the polynomial \(\varphi(\lambda)\) is irreducible, then \(S_n\) can be either an exact endomorphism or an automorphism.

For the proof, consider the endomorphism \(S_n\) on the torus \(T^k\) (\(k\) is the degree of the polynomial \(\varphi(\lambda)\)). If \(S_n\) is not an automorphism on \(T^k\), then the preimage of \(0\) consists of several elements. The closure of the preimages of \(0\) under all \(S_n^p\) (\(p=1,2,\ldots\)) is a torus \(T^{k_1}\) invariant with respect to \(S_n\). If it does not coincide with \(T^k\), then the characteristic polynomial of \(\widetilde S_n\) on \(E^{k_1}\) divides \(\varphi(\lambda)\), which is impossible by virtue of its irreducibility. Thus we obtain \(T^k=T^{k_1}\), i.e., the preimages of \(0\) are dense in the torus \(T^k\), but then they are dense also in \(T^n\); hence it already follows easily that the endomorphism \(S_n\) is exact.

Corollary. An arbitrary endomorphism is the direct product of an exact endomorphism and an automorphism.

Theorem 3. If the characteristic polynomial of an ergodic automorphism \(S_n\) of the torus \(T^n\) has the form \(p(\lambda)=\varphi^s(\lambda)\), where \(\varphi(\lambda)\) is an irreducible polynomial, then \(S_n\) is a Kolmogorov automorphism.

We indicate the main points of the proof. From the ergodicity of the automorphism it follows that there exists a root \(\lambda_1\) of the polynomial \(p(\lambda)\) whose modulus is less than one. Otherwise all roots would have modulus one and the automorphism \(S_n\) would be nonergodic (see \((^3,_8)\)).

Let the vectors \(\tilde{x}_1,\ldots,\tilde{x}_k\) be eigenvectors for the transformation \(\tilde{S}_n\) with eigenvalue \(\lambda_1\), and let the vectors \(\tilde{x}_{k+1},\ldots,\tilde{x}_s\) satisfy the equation

\[ \tilde{S}_n\tilde{x}_i=\lambda_1\tilde{x}_i+\tilde{x}_{i-1}. \]

Represent the torus \(T^n\) as the Euclidean space \(E^n\), in which points whose coordinates differ by an integer are identified. Denote by \(\tilde{\xi}_1\) the partition of \(E^n\) into hyperplanes parallel to the linear subspace generated by the vectors \(\tilde{x}_1,\ldots,\tilde{x}_s\), and by \(\tilde{\xi}_2\) the partition into unit cubes with vertices at integral points. Denote the product of these partitions by \(\tilde{\xi}_3\), and construct the partition \(\tilde{\xi}=\tilde{\xi}_3 \vee \tilde{S}^{-1}\tilde{\xi}_3 \vee \ldots\). Denote by \(\xi\) the image of the partition \(\tilde{\xi}\) in the torus \(T^n\). It turns out that \(\xi\) has the following properties: 1) \(S_n^k\xi < S_n^{k+1}\xi\); 2) \(\bigvee_{k=0}^{\infty} S_n^k\xi=\varepsilon\); 3) \(\bigwedge_{k=0}^{-\infty} S_n^k\xi=\nu\), i.e. \(S_n\) is a Kolmogorov automorphism.

Corollary. Every ergodic automorphism of the torus is a Kolmogorov automorphism.

This follows from the decomposition theorem and from the preceding theorem.

In conclusion we formulate a theorem which, in a special case, was proved by Ya. G. Sinai \((^6)\).

Theorem 4. The entropy of an ergodic automorphism of the torus is equal to

\[ \sum_{|\lambda_i|>1}\log|\lambda_i|. \]

Received
13 I 1961

CITED LITERATURE

\(^1\) A. N. Kolmogorov, DAN, 119, No. 5 (1958).
\(^2\) V. A. Rokhlin, UMN, 4, 2 (30) (1949).
\(^3\) V. A. Rokhlin, Izv. AN SSSR, ser. matem., 13, 329 (1949).
\(^4\) V. A. Rokhlin, UMN, 15, 4 (1960).
\(^5\) V. A. Rokhlin, Izv. AN SSSR, ser. matem., No. 3 (1961).
\(^6\) Ya. G. Sinai, DAN, 124, No. 4 (1959).
\(^7\) B. M. Levitan, Almost-periodic functions, Moscow, 1953.
\(^8\) E. Hecke, Theory of algebraic numbers, 1939.