Past exam of the mathematics course of the University of Cambridge / 2001 / iii / Paper 55 / 1

Ignore the zero-area boundary sets where a reciprocal or its digit is undefined, and put $a=\lfloor 1/x\rfloor$. On this digit branch the natural extension of the continued-fraction Gauss map has inverseThe images are the disjoint horizontal strips $1/(a+1)<y^{\prime }<1/a$, covering the square almost everywhere. Thus the branch calculation proves global invariance without an extra sum over overlapping branches.

Its Jacobian determinant has magnitude $J=[x^2(a+y{)}^2{]}^{-1}$. Sincethe transformed density satisfiesThe change of variables formula therefore proves invariance. Normalization follows fromProjecting onto the first coordinate commutes with the Gauss continued-fraction map, so its invariant marginal is the Gauss measure:One may also check it directly. The inverse branches $h_a(x)=(a+x{)}^{-1}$ givewhere the last sum telescopes. Values assigned at the exceptional boundary points do not affect the absolutely continuous invariant measure.

Use normalized Haar measure $m$ on the real torus. The integer matrix $A$ and its integer inverse induce a toral automorphism; its determinant one preserves $m$. For $k\in {\mathbb{Z}}^n$, let $e_k(x)=e^{2\pi ik\cdot x}$. These characters of a real torus are an orthonormal basis, andSuppose no eigenvalue of $A$ is a root of unity. For $k\ne 0$, the frequency $(A^T{)}^{j}k$ can equal a fixed $\ell$ at most once. Otherwise two such times would give $(A^T{)}^{q}k=k$ for some $q>0$, forcing $A^q-I$ to be singular and an eigenvalue of $A$ to have $q$th power one. Hence the displayed correlation eventually vanishes whenever a nonconstant character is involved. Constant characters give the product of their means.

Linearity gives mixing for finite Fourier series. To pass to arbitrary $f,g\in L^2(m)$, approximate them by trigonometric polynomials. Composition with $T_A^j$ is an isometry of $L^2(m)$, so the Cauchy-Schwarz inequality bounds the approximation errors uniformly in $j$. Let those errors tend to zero after taking the large-$j$ limit. Thus $\int (f\circ T_A^j)\overline{g}dm\to \int fdm\int \overline{g}dm$, which is the required mixing measure-preserving transformation property.

Conversely, if an eigenvalue is a root of unity, choose $q>0$ with $\det [(A^T{)}^q-I]=0$. This is an integer matrix, so its nullspace has a nonzero rational vector; clearing denominators gives $k\in {\mathbb{Z}}^n\setminus \{0\}$ with $(A^T{)}^{q}k=k$. The nonconstant character $e_k$ has mean zero, butfor every $j$. It cannot satisfy mixing. ThereforeHyperbolic toral automorphisms are mixing, but hyperbolicity is not necessary: this proof excludes periodic integer frequencies, not every eigenvalue of modulus one.