Fourier series

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We have seen that a measure-preserving transformation $T$ on a probability space $(X,\mathcal{B},\mu )$ induces an isometry of the space ${\mathsf{L}}^{\mathsf{2}}(X,\mathcal{B},\mu )$ by composition. That is, if $f$ belongs to ${\mathsf{L}}^{\mathsf{2}}(X,\mathcal{B},\mu )$ then so too does $$Tf=f\circ T$$ and moreover $||Tf|{|}_{\mathsf{2}}=||f|{|}_{\mathsf{2}}$.

We restrict now our attention to the special case where our probability space is $[0,1)$ equipped with Lebesgue measure $\lambda$. By paying the small price of working with complex instead of real-valued functions, we will produce an orthonormal basis of ${\mathsf{L}}^{\mathsf{2}}(X,\mathcal{B},\mu )$ that will allow us to more easily verify dynamical properties of measure-preserving transformations on the unit interval.

An orthonormal basis

Write ${\mathsf{L}}^{\mathsf{2}\mathsf{,}\mathbb{C}}([0,1))$ for the collection of functions $f:[0,1)\to \mathbb{C}$ that are Borel measurable and for which $$\int |f{|}^2\mathsf{d}\lambda <\mathrm{\infty }$$ where we identify two complex-valued functions $f,g$ if $$\int |f-g{|}^2\mathsf{d}\lambda$$ holds. Next, define for each $k\in \mathbb{Z}$ the function $${\chi }_k(x)=e^{2\pi ikx}=\cos (2\pi kx)+i\sin (2\pi kx)$$ for each $x\in [0,1)$. Since $|{\chi }_k|=1$ for each $k\in \mathbb{Z}$ and $\lambda ([0,1))=1$ each of the functions ${\chi }_k$ belongs to ${\mathsf{L}}^{\mathsf{2}\mathsf{,}\mathbb{C}}([0,1))$.

In dealing with complex-valued functions we must modify slightly the inner product we work with. Define $$⟨f,g⟩=\int f\cdot \bar{g}\mathsf{d}\lambda$$ whenever $f,g$ belong to ${\mathsf{L}}^{\mathsf{2}\mathsf{,}\mathbb{C}}([0,1))$. With respect to this inner product we have $$⟨{\chi }_k,{\chi }_j⟩=\int {\chi }_k\cdot \overline{{\chi }_j}\mathsf{d}\lambda =\underset{0}{\overset{1}{\int }}{e}^{2\pi i(k-j)x}\mathsf{d}x=\{\begin{array}{ll}1& j=k\\ 0& j\ne k\end{array}$$ which is to say that the collection $\{{\chi }_k:k\in \mathbb{Z}\}$ of functions is an orthonormal system. In fact it is a basis of ${\mathsf{L}}^{\mathsf{2}\mathsf{,}\mathbb{C}}([0,1))$ in the sense that $$f=\sum _{n=1}^{\mathrm{\infty }}⟨f,{\chi }_k⟩{\chi }_k$$ for every $f$ in ${\mathsf{L}}^{\mathsf{2}\mathsf{,}\mathbb{C}}([0,1))$. Note that this is distinct from the meaning of basis in linear algebra: we are not asserting that every $f$ is a finite linear combination of functions from our orthonormal family; instead we are using the norm to say that $$\underset{N\to \mathrm{\infty }}{lim}{\Vert f-\sum _{n=-N}^N⟨f,{\chi }_n⟩{\chi }_n\Vert }_{\mathsf{2}}=0$$ for every $f$ in ${\mathsf{L}}^{\mathsf{2}\mathsf{,}\mathbb{C}}([0,1))$.

Theorem

The ${\chi }_k$ form a basis of ${\mathsf{L}}^{\mathsf{2}\mathsf{,}\mathbb{C}}([0,1))$.

Proof:

We will take this for granted. The key ingredients in the proof are Bessel's inequality, the fact the continuous functions are dense in ${\mathsf{L}}^{\mathsf{2}\mathsf{,}\mathbb{C}}([0,1))$, and the fact that every continuous function on $[0,1)$ is a uniform limit of linear combinations of our orthonormal system. ▮

Ergodicity via Fourier series

Proposition

Every irrational rotation is ergodic.

Proof:

Write $T(x)=x+\alpha mod1$ for some irrational $\alpha$. Fix $f$ in ${\mathsf{L}}^{\mathsf{2}\mathsf{,}\mathbb{C}}([0,1))$ that is $T$ invariant. We can write $$f=\sum _{n\in \mathbb{Z}}c(n){\chi }_n$$ and calculate that $$Tf=\sum _{n\in \mathbb{Z}}c(n)T{\chi }_n=\sum _{n\in \mathbb{Z}}c(n)e^{2\pi in\alpha }{\chi }_n$$ and conclude that $c(n)=c(n)e^{2\pi in\alpha }$ for all $n\in \mathbb{Z}$. As $\alpha$ is irrational we must have $c(n)=0$ for all non-zero $n$. Thus $f=c(0)$ is constant. ▮

Proposition

The map $T(x)=2xmod1$ on $[0,1)$ is ergodic.

Proof:

Fix $f$ in ${\mathsf{L}}^{\mathsf{2}\mathsf{,}\mathbb{C}}([0,1))$ that is $T$ invariant. We can write $$f=\sum _{n\in \mathbb{Z}}c(n){\chi }_n$$ and calculate that $$Tf=\sum _{n\in \mathbb{Z}}c(n)T{\chi }_n=\sum _{n\in \mathbb{Z}}c(n){\chi }_{2n}$$ and conclude that $c(n)=c(2n)$ for all $n\in \mathbb{Z}$. Calculating $$||f|{|}_{\mathsf{2}}^2=⟨f,f⟩=\sum _{n\in \mathbb{Z}}|c(n){|}^2$$ it is not possible for $c$ to take the same non-zero value infinitely often. Thus $c(n)=0$ for all $n\ne 0$ and $f=c(0)$ is constant. ▮

Discrete spectrum and mixing

For the irrational rotation $T(x)=x+\alpha$ each of the functions ${\chi }_n$ is an eigenfunction. We define the class of dynamical systems with this property.

Definition

A measure-preserving transformation $T$ on a probability space $(X,\mathcal{B},\mu )$ has discrete spectrum when ${\mathsf{L}}^{\mathsf{2}\mathsf{,}\mathbb{C}}(X,\mathcal{B},\mu )$ has an orthonormal basis consisting of eigenfunctions.

Ergodic transformations with discrete spectrum can be modelled by systems that look like irrational rotations.

Theorem

If a measure-preserving transformation $T$ is ergodic and has discrete spectrum then it is isomorphic to a measure-preserving transformation $S$ on a compact, Abelian group defined by $S(g)=g+a$ for some $a$ in $G$ with the property that $\{na:n\in \mathbb{Z}\}$ is dense.

Can the doubling map $T(x)=2xmod1$ be modelled by an irrational rotation? Perhaps it is, in disguise, a system with discrete spectrum. In fact, the doubling map is mixing and systems that are ergodic and mixing cannot have non-constant eigenfunctions.

Proposition

The doubling map $T(x)=2xmod1$ on $[0,1)$ is mixing.

Proof:

Fix $f,g$ in ${\mathsf{L}}^{\mathsf{2}\mathsf{,}\mathbb{C}}([0,1))$. Write $$f=\sum _{k\in \mathbb{Z}}c(k){\chi }_{k}g=\sum _{s\in \mathbb{Z}}d(s){\chi }_s$$ and fix $ϵ>0$. There are $K,M\in \mathbb{N}$ such that $${\Vert f-\sum _{|k|\le K}c(k){\chi }_k\Vert }_{\mathsf{2}}<ϵ{\Vert g-\sum _{|m|\le M}d(m){\chi }_m\Vert }_{\mathsf{2}}<ϵ$$ both hold. Write $f_K$ and $g_M$ respectively for the truncated sums above. Now $$|⟨f,T^{n}g⟩-⟨f_M,T^n{g}_K⟩|<ϵ(||f|{|}_{\mathsf{2}}+||g|{|}_{\mathsf{2}})$$ for all $n\in \mathbb{N}$ using the Hölder inequality. Choose $N$ with $2^N>M$. For all $n\ge N$ we have $$⟨f_M,T^n{g}_K⟩=\sum _{|m|\le M}\sum _{|k|\le K}c(m)\overline{d(k)}⟨{\psi }_m,{\psi }_{2^{n}k}⟩=c(0)\overline{d(0)}$$ because $2^{n}k=m$ only happens when $m=0$ and $k=0$. Thus $$|⟨f,T^{n}g⟩-c(0)\overline{d(0)}|<ϵ(||f|{|}_{\mathsf{2}}+||g|{|}_{\mathsf{2}})$$ for all $n\ge N$ and we are done. ▮

We can now distinguish the irrational rotation and the doubling map using the Koopman operator. Indeed, suppose that $f$ is an eigenfunction for the doubling map $T$ with eigenvalue $\eta$. Then $$|⟨f,T^{n}f⟩|=|⟨f,{\eta }^{n}f⟩|=|\overline{{\eta }^n}|||f|{|}_{\mathsf{2}}^2=||f|{|}_{\mathsf{2}}^2$$ whereas mixing gives $$⟨f,T^{n}f⟩\to ⟨f,1⟩⟨1,f⟩=0$$ if $\eta \ne 1$ because eigenfunctions with distinct eigenvalues are orthogonal. Thus $||f|{|}_{\mathsf{2}}=0$ which is not possible and $T$ has no eigenfunctions.