Abstract Generated abstract
The paper proves that, in the weak topology on the space of invertible measure-preserving transformations of the unit interval, the automorphisms without simple spectrum form a set of first category. The argument constructs dense \(G_\delta\) classes of transformations approximable along subsequences by cyclic dyadic permutations with rapidly decreasing error, then shows that such approximation implies the existence of a cyclic generating element in \(L^2\). Combined with the known genericity of continuous spectrum, the result yields an everywhere dense \(G_\delta\) set of automorphisms with simple continuous spectrum.
Full Text
UDC 519.53
MATHEMATICS
S. A. Yuzvinsky
ON METRIC AUTOMORPHISMS WITH SIMPLE SPECTRUM
(Presented by Academician A. N. Kolmogorov on 14 IV 1966)
Let \(M\) be the unit interval with the usual Lebesgue measure \(\mu\); let \(\mathfrak A\) be the set of all automorphisms (invertible measure-preserving transformations) of the space \(M\), endowed with the weak topology (see \(\left({}^{2}\right)\)), and let \(\mathfrak P\) be the set of automorphisms in \(\mathfrak A\) having simple spectrum (see \(\left({}^{1}\right)\)). The purpose of the present note is to prove the following theorem.
Theorem 1. In the space \(\mathfrak A\), the complement of the set \(\mathfrak P\) is of first category.
From this theorem and from the well-known theorem on the massiveness of the set of automorphisms with continuous spectrum in \(\mathfrak A\) (see \(\left({}^{1}\right)\)) it follows that in \(\mathfrak A\) there exists an everywhere dense \(G_\delta\) consisting of automorphisms with simple continuous spectrum.
History of the question. Even before the war A. N. Kolmogorov indicated an example of an automorphism for which one could suppose that it has simple spectrum. A proof of the simplicity of the spectrum in this example was found in 1958 by I. V. Girsanov \(\left({}^{3}\right)\). This was an example of a probability-theoretic origin. In September 1965, at one of the sessions of the Khumsan school on ergodic theory, A. B. Katok, A. G. Kushnirenko, V. I. Oseledets, and A. M. Stepin presented simpler and more geometric examples of automorphisms with simple and finite-multiple nondiscrete spectrum, together with methods for estimating the multiplicity of the spectrum. In November 1965, at his seminar on ergodic theory, V. A. Rokhlin expressed the supposition that a suitable generalization of these methods, together with more old-fashioned tools, would make it possible to prove Theorem 1. The present work is an implementation of this program.
- We shall call dyadic intervals of rank \(n\) \((n=1,2,\ldots)\) the intervals
\(C_n^i=(i/2^n,(i+1)/2^n)\) \((i=0,1,\ldots,2^n-1)\), and dyadic permutations of rank \(n\) automorphisms of the space \(M\) cyclically permuting these intervals. For an arbitrary positive sequence \(f(n)\), denote by \(\mathfrak B_f\) the set of automorphisms \(T\in\mathfrak A\) such that \(T\in\mathfrak B_f\) if and only if, for some strictly increasing sequence of natural numbers \(q_n\) and a sequence of dyadic permutations \(S_n\) of rank \(q_n\),
\[ \mu\bigl(TC_{q_n}^i \Delta S_n C_{q_n}^i\bigr)<f(q_n), \qquad i=0,1,\ldots,2^{q_n}-1;\quad n=1,2,\ldots \]
(\(\Delta\) denotes the symmetric difference).
We shall prove that for any sequence \(f\) the set \(\mathfrak B_f\) is an everywhere dense \(G_\delta\) in \(\mathfrak A\). To this end denote by \(\mathfrak B_f^m\) the set of automorphisms \(T\in\mathfrak A\) such that \(T\in\mathfrak B_f^m\) if and only if, for some dyadic permutation \(S_m\) of rank \(m\),
\[ \mu\bigl(TC_m^i \Delta S_m C_m^i\bigr)<f(m), \qquad i=0,1,\ldots,2^m-1. \]
It is clear that the set \(\mathfrak B_f^m\) is open in \(\mathfrak A\) and
\[ \mathfrak B_f=\bigcap_{l=1}^{\infty}\bigcup_{m=l}^{\infty}\mathfrak B_f^m . \]
Consequently, \(\mathfrak B_f\) is \(G_\delta\). Next, let \(\alpha\) be an irrational number for which there exists a sequence of irreducible fractions \(\alpha_n=p_n/2^{k_n}\) with strictly increasing denominators such that \(|\alpha-\alpha_n|<\frac12 f(k_n)\). Denote by \(A\) the shift mod \(1\) by \(\alpha\), and by \(A_n\) the shift mod \(1\) by \(\alpha_n\). If \(Q\) is an arbitrary dyadic permutation of some rank \(s\) and \(n\) is such that \(k_n>s\), then \(QC_{k_n}^i\) \((i=0,\ldots,2^{k_n}-1)\) is a dyadic interval of rank \(k_n\), the automorphism \(Q^{-1}A_nQ\) is a dyadic permutation of rank \(k_n\), and
\[ \mu\bigl(Q^{-1}AQC_{k_n}^i \,\Delta\, Q^{-1}A_nQC_{k_n}^i\bigr)<f(k_n),\qquad i=0,1,\ldots,2^{k_n}-1. \]
Consequently, \(Q^{-1}AQ\in\mathfrak B_f\). Since in the space \(\mathfrak A\) the set of all automorphisms conjugate to the automorphism \(A\) is everywhere dense \(\bigl((2),\text{ p. }108\bigr)\), and the set of dyadic permutations is also everywhere dense \(\bigl((2),\text{ p. }92\bigr)\), it follows that the set \(\mathfrak B_f\) is everywhere dense.
- We now show that if, for the sequence \(f\), the condition
\[ f(n)=o\bigl[2^{-n(5+2^n)}\bigr], \]
is satisfied, then \(\mathfrak B_f\subset\mathfrak P\).
Let \(T\in\mathfrak B_f\) and let \(\{q_n\}, \{S_n\}\) be the corresponding sequences. Suppose, moreover, that \(\beta\) is a real number \((0<\beta<1)\) for which there exists a sequence of irreducible rational fractions \(\beta_n=t_n/r_n\), where \(r_n=2^{q_{k_n}}\), \(k_n\to\infty\), such that
\[ |\beta-\beta_n|<(r_n)^{-(r_n+4)}. \tag{1} \]
Put \(R_n=S_{k_n}\), and denote by \(U\) the operator in \(L^2(M)\) conjugate to \(T\), by \(V_n\) the operator conjugate to \(R_n\), and by \(\chi,\chi_n\), and \(\widetilde{\chi}_n\) the characteristic functions of the intervals \([0,\beta)\), \([0,\beta_n)\), and \([0,1/r_n)\). We shall prove that the function \(\chi\) is a generating element for \(U\), i.e. that the linear span of the set \(\{U^i\chi\}\) \((i=0,1,\ldots)\) is everywhere dense in \(L^2(M)\).
First note that for any \(n\) the function \(\chi_n\) is a generating element for \(V_n\) in the subspace \(L_n\) of functions that are constant on dyadic intervals of rank \(q_{k_n}\). Indeed, otherwise in \(L_n\) there would be an eigenfunction of the operator \(V_n\) orthogonal to \(\chi_n\), i.e., in other words, the equality \(\sum_{i=1}^{p}\lambda_i=0\) would hold, where \(\lambda_i\) are roots of unity of degree \(r_n\) and \(p\) is an odd number \((1\le p<r_n)\). But it is not difficult to show by induction on \(n\) that this equality is impossible.
Now let \(g\) be a bounded function from \(L^2(M)\), \(|g(x)|<K\) \((x\in M)\), and let \(\varepsilon\) be an arbitrary positive number. Since \(r_n\to\infty\) and \(R_n\) cyclically permutes dyadic intervals of rank \(q_{k_n}\), there exists a natural number \(N\) such that for \(n>N\)
\[ \left\|g-\sum_{i=1}^{r_n}c_iV_n^i\widetilde{\chi}_n\right\|<\varepsilon, \tag{2} \]
where \(c_i\) are some real numbers depending on \(n\), and \(\sum |c_i|<K\). Fix some number \(n>N\) for which
\[ 2K\sqrt{f(q_{k_n})}\,(r_n)^{(r_n+5)/2}<\varepsilon, \tag{3} \]
\[ 2K/r_n<\varepsilon. \tag{4} \]
Since \(\widetilde{\chi}_n\) is a generating element for \(V_n\) in \(L_n\), in particular,
\[ \widetilde{\chi}_n=\sum_j d_j V_n^j\chi_n, \]
where, according to Hadamard’s inequality, \(|d_j|<(r_n)^{r_n/2}\). Thus—
it follows that
\[ \left\|g-\sum_i c_i V_n^i \tilde{\chi}_n\right\| = \left\|g-\sum_i c_i V_n^i \chi_n\right\|, \tag{5} \]
where
\[ \tilde{c}_i=\sum_{\substack{k+l=i\\(\operatorname{mod} m)}} c_k d_l, \]
and therefore
\[ |\tilde{c}_i|<2K(r_n)r_n/2. \tag{6} \]
Applying the triangle inequality, we obtain
\[ \left\|g-\sum_i \tilde{c}_i U^i\chi\right\| \leq \left\|g-\sum_i \tilde{c}_i V_n^i\chi_n\right\| +\sum_i |\tilde{c}_i|\,\|V_n^i\chi_n-U^i\chi_n\| +\sum_i |\tilde{c}_i|\,\|\chi_n-\chi\|. \]
We estimate each of the three terms on the right-hand side. According to (2) and (5),
\[ \left\|g-\sum_i \tilde{c}_i V_n^i\chi_n\right\|<\varepsilon. \tag{7} \]
Put
\[ A=\sum_i |\tilde{c}_i|\,\|V_n^i\chi_n-U^i\chi_n\| \]
and
\[ B=\sum_i |\tilde{c}_i|\,\|\chi_n-\chi\|. \]
Since
\[ A\leq \sum_i |\tilde{c}_i|\sum_{j=0}^{i-1}\|V_n V_n^j\chi_n-UV_n^j\chi_n\| = \sum_i |\tilde{c}_i|\sum_{j=0}^{i-1}\sqrt{\mu(R_n E_n^j\Delta T E_n^j)}, \]
where \(E_n^j\) are certain subsets of the space \(M\) consisting of dyadic intervals of rank \(q_{k_n}\), it follows from (6) and the properties of the sequence \(\{R_n\}\) that
\[ A\leq 2K(r_n)(r_n+5)/2\sqrt{f(q_{k_n})}, \]
and, according to (3),
\[ A<\varepsilon. \tag{8} \]
Finally,
\[ B=\sqrt{|\beta-\beta_n|}\sum_i |\tilde{c}_i|, \]
and from (1), (6), and (4) it follows that
\[ B<2K/r_n<\varepsilon. \tag{9} \]
From (7), (8), and (9) it follows that
\[ \left\|g-\sum_i \tilde{c}_i U^i\chi\right\|<3\varepsilon. \tag{10} \]
In view of the arbitrariness of the choice of \(\varepsilon\) and \(g\), inequality (10) means that the linear span of the set \(\{U^i\chi\}\) \((i=0,1,\ldots)\) is everywhere dense in the set of bounded functions in \(L^2(M)\), and consequently everywhere dense in \(L^2(M)\).
Leningrad State Pedagogical Institute
named after A. I. Herzen
Received
9 IV 1966
REFERENCES
- V. A. Rokhlin, Uspekhi Mat. Nauk, 4, no. 2 (1949).
- P. R. Halmos, Lectures on Ergodic Theory, IL, 1959.
- I. V. Girsanov, Dokl. Akad. Nauk SSSR, 119, no. 5 (1958).