V. V. Ryzhikov, “Kolmogorov and Rokhlin spectral problems in the class of mixing automorphisms”, Russian Math. Surveys, 79:6 (2024), 1093

The Komogorov problem of the group property of the spectra of automorphisms of the Lebesgue space is well known in ergodic theory. Komogorov conjectured that the maximal spectral type of an ergodic automorphism dominates its convolution. Sinai [1] verified the conjecture for automorphisms satisfying certain strong mixing conditions. For non-mixing automorphisms the conjecture was refuted, and counterexamples were presented by Katok and Stepin [2] and Oseletets [3]. For mixing automorphisms the question of the group property of their spectra had remained open for a long time. The answer owes much to a solution of another problem, which is known in ergodic theory as the Rokhlin problem of the homogeneous spectrum of an ergodic automorphism (see Anosov’s pamphlet [4] on this topic). In [5] we proved that there exists a mixing automorphism

$T$ such that the product

$T\times T$ has homogeneous spectrum of multiplicity 2. The spectral measure of such an automorphism

$T$ is mutually singular with its convolution square. Thus, along with the Rokhlin problem, we solved the Kolmogorov problem in the class of mixing automorphisms. However, the method of approximating mixing actions by non-mixing ones, which we used in [5], produces no explicit examples. Explicit constructions were described in [6], namely, staircase constructions with logarithmic growth of parameters. In this note we improve that result to a power growth of parameters.

The staircase construction. Let a sequence of integers

$r_j\to\infty$ ,

$r_j>2$ , be fixed, and let

$h_1=10$ . To these parameters we will assign an automorphism

$T$ of the Lebesgue space, which is called a staircase construction (see [6]–[8]). The authormorphism is constructed in several steps, starting with

$j=1$ . At each step

$j$ we have a system of disjoint intervals

$E_j TE_j,T^2E_j,\dots, T^{h_j-1}E_j$ (the tower at step

$j$ ). Let

$X_j$ denote the union of these intervals. By definition the transformation

$T$ acts as a usual translation of these intervals (although it has not yet been defined on

$T^{h_j-1}E_j$ ). We describe the transition from step

$j$ to step

$j+1$ . We partition

$E_j$ into

$r_j$ intervals

$E_j^1,E_j^2,\dots,E_j^{r_j}$ of equal length and consider the intervals

$E_j^i, TE_j^i,\dots,T^{h_j+i}E_j^i$ for

$i=1,2,\dots,r_j-1$ and

$E_j^{r_j},TE_j^{r_j},\dots,T^{h_j-1}E_j^{r_j}$ (all of which are pairwise disjoint). Putting

$E_{j+1}=E^1_j$ , for all

$i<r_j$ we set

$TT^{h_j+i}E_j^i=E_j^{i+1}$ . This produces the tower at step

$j+1$ , which consists of

$E_{j+1},TE_{j+1},\dots,T^{h_{j+1}-1}E_{j+1}$ , where

$h_{j+1}=h_j+\sum_{i=1}^{r_j-1}(h_j+i)$ . Note that the (partial) definition of

$T$ at step

$j$ is preserved at the subsequent steps. Continuing without limit we obtain an invertible transformation

$T\colon X\to X$ preserving the standard Lebesgue measure on

$X=\bigcup_j X_j$ . Such a transformation is known to have spectrum of multiplicity one and mixing property [7].

notfound Scheme of the proof. The characteristic function

$f$ of the interval

$E_1$ is known to be a cyclic vector of the operator

$\mathbf T$ corresponding to

$T$ . We prove that the vectors

${\mathbf T}^pf\otimes f+f\otimes {\mathbf T}^pf$ belong to the cyclic space

$C_{f\otimes f}$ of the operator

${\mathbf T}\otimes {\mathbf T}$ . This shows that the symmetric tensor power

${\mathbf T}\odot {\mathbf T}$ has spectrum of multiplicity one. Set

${\mathbf Q}_n=(1/n)\sum_{i=0}^{n-1}{\mathbf T}^{-i}$ , and for

$n>2$ let

$\begin{equation*} \Delta_{n}=n^2{\mathbf Q}_{n}\otimes {\mathbf Q}_{n}- 2(n-1)^2{\mathbf Q}_{n-1}{\mathbf T}\otimes {\mathbf Q}_{n-1}{\mathbf T}+ (n-2)^2{\mathbf Q}_{n-2}{\mathbf T}^2\otimes {\mathbf Q}_{n-2}{\mathbf T}^2. \end{equation*} \notag$

Note that for

$0<p<n$ we have

$\begin{equation} {\mathbf T}^{p}f\otimes f+f\otimes {\mathbf T}^{p}f={\mathbf T}^{p-n}\otimes {\mathbf T}^{p-n}(\Delta_{n-p}\Delta_{n}-\Delta_{2n-p})f\otimes f. \end{equation} \tag{1}$

Let

$J_n=\{j\colon r_j=n\}$ ,

$|J_n|<\infty$ and

$|J_n|/n^{8}\to \infty$ . Taking account of (1), the easy inequality

$\|\Delta_{n-p}\Delta_{n}\|< 16 n^4$ , and the relation

$\begin{equation*} \biggl\|\frac{1}{|J_n|}\displaystyle\sum_{j\in J_n}{\mathbf T}^{h_j}f\otimes {\mathbf T}^{h_j}f-{\mathbf Q}_n f\otimes {\mathbf Q}_n f\biggr\|^2< \frac{C}{|J_n|} \end{equation*} \notag$

for some constant

$C$ , we see that the distance of

${\mathbf T}^pf\otimes f+f\otimes {\mathbf T}^pf$ to the cyclic space

$C_{f\otimes f}$ is zero. The condition

$|J_n|/n^8\to\infty$ holds for

$r_j=[10\, j^d]$ ,

$1-d>8d>0$ , because then

$|J_n|\sim n^{1-d}$ . Hence for

$0<9d<1$ we have

${\mathbf T}^pf\otimes f+f\otimes {\mathbf T}^pf\in C_{f\otimes f}$ for all

$p>0$ (if

$p<0$ , then the argument is similar). Thus, the spectrum of

${\mathbf T}\odot {\mathbf T}$ has multiplicity one, which implies that the spectrum of

$T$ has no group property, and the product

${T}\times {T}$ has homogeneous spectrum of multiplicity two.

$\Box$

Note that Tikhonov [9] proved the existence of mixing automorphisms with homogeneous spectrum of multiplicity

$m>2$ . No constructive examples explicitly realizing such multiplicities are known so far.

Citation in format AMSBIB

\Bibitem{Ryz24}

\by V.~V.~Ryzhikov

\paper Kolmogorov and Rokhlin spectral problems in the class of mixing automorphisms

\jour Russian Math. Surveys

\yr 2024

\vol 79

\issue 6

\pages 1093--1094

\mathnet{http://mi.mathnet.ru/eng/rm10220}

\crossref{https://doi.org/10.4213/rm10220e}

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