Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 358 / 3

The map $F$ is a nonsingular transformation with respect to $\omega$ whenEquivalently, the pushforward measure $F_{\ast }\omega (B)=\omega (F^{-1}(B))$ satisfies $F_{\ast }\omega \ll \omega$.

For an essentially bounded observable $g$, define the Koopman operatorNonsingularity makes this well defined on almost-everywhere equivalence classes. By the Radon-Nikodym theorem,Consequently the bounded Koopman operator criterion is

The map $F$ is a measure-preserving transformation whenfor every Borel set $B$, equivalently $F_{\ast }\omega =\omega$. In this case its Koopman operator is an isometry on $L^2(\omega )$.

The map is invertible with respect to $\omega$ when there is a measurable $G:X\to X$ such thatalmost everywhere. For an invertible measure-preserving system, $K_G=K_F^{-1}=K_F^{\ast }$, so its Koopman operator is unitary.

Suppose first that the system is ergodic and $K_{F}g=g$. For each real $t$, the level setis invariant modulo a null set, so $\omega (E_t)\in \{0,1\}$. The distribution function of $\mathrm{Re}g$ can therefore jump only once, which makes $\mathrm{Re}g$ constant almost everywhere. The same argument applies to $\mathrm{Im}g$.

Conversely, if $E$ is invariant, then $K_F{1}_E=1_E$. If every invariant $L^2$ function is constant, the indicator function $1_E$ is almost everywhere zero or one, and hence $\omega (E)=0$ or $1$. This proves the invariant-function characterization of ergodicity.

Use the Fourier basisof $L^2([-\pi ,\pi {]}_{\mathrm{per}})$. For the rotation $F(x)=x+a$,If $a/\pi$ is irrational, $e^{ika}=1$ implies $k=0$. Hence every fixed $L^2$ function has only its constant Fourier coefficient, and part (i) proves ergodicity.

If $a/\pi =p/q$ is rational, thenis a nonconstant fixed function because $e^{i2qa}=e^{i2\pi p}=1$. Part (i) now shows that the system is not ergodic. Therefore the ergodicity criterion for a circle rotation is

For inexact information in the SCI hierarchy, replace every exact evaluation $\lambda \in \Lambda$ by a family of admissible approximations ${\lambda }_m$ satisfyingAn algorithm must converge for every admissible choice of approximations, not merely for one favored encoding.

For continuous nonsingular maps $F:X\to X$, take the evaluations to be arbitrary point queries. At precision $m$, a query at $x\in X$ returns any $y$ satisfyingThus the information set contains all triples $(x,m,y)$ satisfying this inequality. This is a perfect measurement device for a dynamical system: it can sample any state, at any requested accuracy, with no fixed noise floor. The finite-information rule still requires each terminating computation to make only finitely many such measurements.

Assume for contradiction that a sequence of general algorithms ${\Gamma }_n$ decides ergodicity from the perfect measurement data, so that ${\Gamma }_n(F)$ eventually equals ${\Xi }_{\mathrm{erg}}(F)$ for every $F\in \Omega$.

Restrict the input class to the circle rotationsFrom inexact information in the SCI hierarchy for the real number $a$, one can answer every requested measurement of $F_a(x)$ to the same precision. The supposed algorithms would therefore give a one-limit decision procedure forbecause part (b)(ii) identifies ergodicity with irrationality.

Every finite-information general algorithm is locally constant on a sufficiently small cylinder of the inexact data. A pointwise limit of a sequence of such functions is a Baire class one function. But the rationality indicator is discontinuous at every real number: every interval contains both rational and irrational numbers. The theorem that the discontinuity set of a Baire class one function is meagre, or directly rationality indicator is not Baire class one, gives a contradiction.

Hence no one-limit tower of general algorithms can decide ergodicity, even with the perfect measurement device: