MATHEMATICS

N. M. AKULINICHEV

ON A DYNAMICAL SYSTEM CONNECTED WITH THE DISTRIBUTION OF THE FRACTIONAL PARTS OF A POLYNOMIAL OF THE SECOND DEGREE

(Presented by Academician I. M. Vinogradov on 17 XI 1961)

A dynamical system connected with the distribution of the fractional parts of a linear function was considered by A. Ya. Khinchin. In (¹), the ergodicity of the dynamical system \((\Omega, T^x, \mu)\) is established by elementary methods, where \(\Omega = [0, 1)\); \(T^x(\alpha) = \{\alpha + x\gamma\}\); \(x = 1, 2, \ldots\); \(\gamma\) is a fixed irrational number; \(\mu\) is Lebesgue measure on \(\Omega\); \(\{\beta\}\) is the fractional part of the number \(\beta\).

We consider a dynamical system connected with the distribution of the fractional parts of a polynomial of the second degree. Let \(\Omega\) be the unit square \(0 \le \alpha < 1,\ 0 \le \beta < 1\); \(\mu\) be Lebesgue measure on \(\Omega\). Define a mapping \(T\) of \(\Omega\) onto itself by the formula

\[ T(\alpha,\beta)=\bigl(\{\alpha+2\beta+\gamma\},\{\beta+\gamma\}\bigr), \]

where \(\gamma\) is a fixed irrational number. It is easily established that for any natural \(x\) and \(y\)

\[ T^x(\alpha,\beta)=\bigl(\{\alpha+2\beta x+\gamma x^2\},\{\beta+\gamma x\}\bigr), \]

\[ T^xT^y(\alpha,\beta)=T^{x+y}(\alpha,\beta). \]

The transformation \(T\) is one-to-one, and \(\mu\) is an invariant measure with respect to \(T\). It is not difficult to write down formulas for the transformation \(T'\) of the dynamical system \((\Omega', T'^{\,x}, \mu')\); \(x = 1, 2, \ldots\), \(\Omega'\) is the unit hypercube \(0 \le \alpha_i < 1\); \(i = 1, 2, \ldots, n\); \(\mu'\) is \(n\)-dimensional Lebesgue measure, connected with the distribution of the fractional parts of a polynomial of the \(n\)-th degree.

Theorem 1. Let \(s\) be a fixed natural number. The dynamical system \((\Omega, T^{sx}, \mu)\), \(x = 1, 2, \ldots\), is ergodic.

Proof. Denote by \(U\) the operator in the space \(L^2_{\Omega,\mu}\) of complex-valued functions integrable over the square with respect to the measure \(\mu\):

\[ Uf(\alpha,\beta)=f\bigl(T^s(\alpha,\beta)\bigr). \]

The operator \(U\) is unitary. It is enough to prove that for any function \(f(\alpha,\beta)\) from \(L^2_{\Omega,\mu}\), from the equality

\[ Uf(\alpha,\beta)=f(\alpha,\beta) \]

for almost all points \((\alpha,\beta)\) it follows that for almost all \((\alpha,\beta)\)

\[ f(\alpha,\beta)=C, \]

where \(C=\mathrm{const}\). Let

\[ f(\alpha,\beta)\sim \sum_{m_1,m_2=-\infty}^{\infty} c_{m_1,m_2} e^{2\pi i(m_1\alpha+m_2\beta)} \]

the expansion of \(f(\alpha,\beta)\) in a Fourier series. Making a change of variables in the integrals

\[ c_{m_1,m_2}=\iint_{\Omega} f(\alpha,\beta)e^{-2\pi i(m_1\alpha+m_2\beta)}\,d\mu, \]

we obtain, by virtue of the invariance of the measure \(\mu\), the relations

\[ c_{m_1,m_2}=e^{-2\pi i\gamma(m_1s^2+m_2s)}c_{m_1,\,2sm_1+m_2}. \]

Suppose there exists a pair of integers \(m_1,m_2\) with \(m_1=0,\ m_2\ne0,\ c_{m_1,m_2}\ne0\)—this is impossible by the irrationality of \(\gamma\). The case in which in the expansion of \(f(\alpha,\beta)\) there exists a nonzero \(c_{m_1,m_2}\) with \(m_1\ne0\) contradicts Bessel’s inequality for \(f(\alpha,\beta)\), since then this expansion contains an infinite number of nonzero coefficients, equal to one another in modulus, of the form \(c_{m_1,\,2ksm_1+m_2}\), \(k=1,2,\ldots\). Consequently, \(f(\alpha,\beta)=c_{0,0}\) for almost all \((\alpha,\beta)\).

Corollary. Let \(f(\alpha,\beta)\in L^1_{\Omega,\mu}\). Then for any natural number \(s\)

\[ \lim_{p\to\infty}\frac1p\sum_{x=0}^{p-1} f\bigl(\{a+2\beta sx+s^2x^2\gamma\},\{\beta+sx\gamma\}\bigr) =\iint_{\Omega} f(\alpha,\beta)\,d\mu . \]

The proof follows from the Birkhoff–Khinchin theorem \(\bigl((^2),\) p. 31\(\bigr)\). Theorem 2 gives information on the question of the existence of a singly mixing transformation that is not doubly mixing (see \((^2)\), p. 133).

Theorem 2. Let \(M\) be the set of functions from \(L^2_{\Omega,\mu}\) that depend only on \(\alpha\):

\[ \varphi(\alpha,\beta)=\varphi(\alpha), \]

with mean value equal to zero. The transformation \(T\) is singly mixing on \(M\) and is not doubly mixing.

Proof. Let \(\varphi_1(\alpha,\beta),\varphi_2(\alpha,\beta)\in M\)

\[ \varphi_1(\alpha,\beta)\sim \sum_{m_1=-\infty}^{\infty} c_{m_1}e^{2\pi i m_1\alpha},\qquad \varphi_2(\alpha,\beta)\sim \sum_{m_2=-\infty}^{\infty}{}' d_{m_2}e^{2\pi i m_2\alpha} \]

\[ \left(\sum_{m=-\infty}^{\infty}{}' \text{ denotes the absence in the sum of the term with zero index}\right) \]

—be their Fourier series, and let \(l>0\) be an integer. We have \((s=1)\)

\[ \iint_{\Omega} U^l\varphi_1(\alpha,\beta)\varphi_2(\alpha,\beta)\,d\mu = \]

\[ =\sum_{m_1=-\infty}^{\infty}{}'\sum_{m_2=-\infty}^{\infty}{}' c_{m_1}d_{m_2}e^{2\pi i m_1l^2\gamma} \iint_{\Omega} e^{2\pi i[(m_1+m_2)\alpha+2m_1l\beta]}\,d\mu=0. \]

Thus, for all \(l\ge1\),

\[ \iint_{\Omega} U^l\varphi_1(\alpha,\beta)\varphi_2(\alpha,\beta)\,d\mu = \iint_{\Omega}\varphi_1(\alpha,\beta)\,d\mu \iint_{\Omega}\varphi_2(\alpha,\beta)\,d\mu . \]

This means precisely that \(T\) is mixing on \(M\).

Next, let \(\varphi_0(\alpha,\beta)=e^{2\pi i\alpha}\), \(\varphi_1(\alpha,\beta)=e^{2\pi i\alpha}\), \(\varphi_2(\alpha,\beta)=e^{-4\pi i\alpha}\). Take the sequence of triples of natural numbers

\[ (k_l^{(0)},k_l^{(1)},k_l^{(2)})=(2l,4l,3l),\qquad l=1,2,\ldots \]

The conditions \(\varphi_i(\alpha,\beta)\in M,\ i=1,2,3,\)

\[ \lim_{l\to\infty}\min_{i<j}\left|k_l^{(i)}-k_l^{(j)}\right|=\infty \]

are satisfied,

\[ \iint_{\Omega} U^{k_l^{(0)}}\varphi_0(\alpha,\beta)\, U^{k_l^{(1)}}\varphi_1(\alpha,\beta)\, U^{k_l^{(2)}}\varphi_2(\alpha,\beta)\,d\mu = e^{2\pi i l^2/2^{\gamma}}, \]

\[ \lim_{l\to\infty} e^{2\pi i l^2/2^{\gamma}} \]

does not exist; on the other hand,

\[ \iint_{\Omega}\varphi_0(\alpha,\beta)\,d\mu\cdot \iint_{\Omega}\varphi_1(\alpha,\beta)\,d\mu\cdot \iint_{\Omega}\varphi_2(\alpha,\beta)\,d\mu =0. \]

Consequently, \(T\) is not doubly mixing on \(M\).

Mathematical Institute named after V. A. Steklov
Academy of Sciences of the USSR

Received
9 XI 1961

References Cited

\(^{1}\) A. Ya. Khinchin, Matem. sborn., 41, 11 (1934).
\(^{2}\) P. Halmos, Lectures on Ergodic Theory, Moscow, 1959.