Posted by: yglima | March 30, 2010

ERT10: Compact Systems

We now turn attention to the structured situation. Consider a measure-preserving system {\mathbb X=(X,\mathcal A,\mu,T)}. Remember that, by ERT8, this is the case when the Koopman-von Neumann operator {T:L^2(\mu)\rightarrow L^2(\mu)} has a basis of eigenvectors. Assume that {\sigma(T)\subset \mathbb S^1} is the multiset of eigenvalues and{f_\lambda} is the eigenvector associated to {\lambda\in\sigma(T)}.

Definition 1 We say that {\mathbb X} is a compact system if\displaystyle \overline{\langle \{f_\lambda\,;\,\lambda\in\sigma(T)\}\rangle}=L^2(\mu).

Let’s investigate an example: given {\alpha\in\mathbb R}, consider the rotation

\displaystyle \begin{array}{rcrcl} R_\alpha&:&\mathbb S^1 &\rightarrow&\mathbb S^1\\ & & x&\mapsto &x+\alpha. \end{array}

If {f_n(x)=e^{2\pi inx}}, {n\in\mathbb Z}, then

\displaystyle \begin{array}{rcl} Tf_n&=&e^{2\pi in(x+\alpha)}\\ & &\\ &=&e^{2\pi in\alpha}\cdot f_n\,, \end{array}

that is, each {f_n} is an eigenvector. We know, by Fourier analysis, that {\{f_n\}_{n\in\mathbb Z}} generates the space {L^2(\mu)}, where {\mu} is the Lebesgue measure on {\mathbb S^1}. This implies that {R_\alpha} is compact. In this setting, it is straightforward to prove multiple recurrence. Assume that {A\subset\mathbb S^1} is an interval and fix {k\in\mathbb N}. We want to show the existence of {n\in\mathbb N} such that

\displaystyle \mu\left(A\cap T^nA\cap\cdots\cap T^{kn}A\right)>0.

This is easy: given {\varepsilon>0}, take {\delta>0} such that

\displaystyle \mu\left(A\cap(A+y)\right)>\mu(A)-\varepsilon/2\,,

for every {y\in(-\delta,\delta)}. This implies that

\displaystyle \int_{\mathbb S^1}|\chi_A-\chi_{A+y}|d\mu<\varepsilon

whenever {|y|<\delta}. If {n\in\mathbb N} is such that {\{n\alpha\}\in(-\delta/k,\delta/k)} {(*)},

\displaystyle \int_{\mathbb S^1}|\chi_A-\chi_{A+jn\alpha}|d\mu<\varepsilon\,,\ \ j=0,1,\ldots,k\,,

which implies that

\displaystyle \begin{array}{rcl} \int_{\mathbb S^1}|\chi_A\chi_{A+n\alpha}\cdots\chi_{A+kn\alpha}-{\chi_A}^{k+1}|d\mu&<&k\varepsilon\\ \Longrightarrow\hspace{2cm} \mu\left(A\cap{R_\alpha}^nA\cap\cdots\cap{R_\alpha}^{kn}A\right)&>&\mu(A)-k\varepsilon. \end{array}

Also, note that the set of {n\in\mathbb N} satisfying {(*)} is syndetic, so that the above argument actually proves that

\displaystyle \liminf_{M-N\rightarrow+\infty}\dfrac{1}{M-N}\sum_{n=N+1}^{M}\mu\left(A\cap{R_\alpha}^nA\cap\cdots\cap{R_\alpha}^{kn}A\right)>0.

The same argument holds for any {A\subset\mathbb S^1} with positive measure. In fact, by Lebesgue density theorem, given {\varepsilon>0}, there exists {\delta=\delta(\varepsilon)} such that

\displaystyle \mu\left(A\cap(A+y)\right)>\mu(A)-\varepsilon/2\,,\ \forall\,y\in(-\delta,\delta),

which is the main inequality used in the above argument. We get our first result.

Theorem 2 If {A\subset\mathbb S^1}, {\mu(A)>0}, then\displaystyle \liminf_{M-N\rightarrow+\infty}\dfrac{1}{M-N}\sum_{n=N+1}^{M}\mu\left(A\cap{R_\alpha}^nA\cap\cdots\cap{R_\alpha}^{kn}A\right)>0

for any {k\in\mathbb N}.

Note that the main tool is that, for various {n}, the functions

\displaystyle \chi_A,\chi_{A+n\alpha},\ldots,\chi_{A+kn\alpha}

are simultaneously close to each other. This is what we are looking for: structure (given by the algebraic structure of {R_\alpha}) implying “almost periodicity”, as will be explained in the next section.

1. Almost periodicity

Consider a mps {\mathbb X=(X,\mathcal A,\mu,T)}.

Definition 3 A function {f\in L^2(\mu)} is almost periodic if the set {\{T^nf;n\in\mathbb Z\}} is pre-compact in {L^2(\mu)}.

In other words, if {\overline{\{T^nf;n\in\mathbb Z\}}} is a compact subset of {L^2(\mu)}, considered with the norm topology.

We will see below that the systems for which every element of {L^2(\mu)} is almost periodic are exactly the compact ones.

At first impression, this does not seen the expected definition we want, but the following proposition clarifies the apparent uncorrelation.

Proposition 4 Given a mps {\mathbb X=(X,\mathcal B,\mu,T)} and {f\in L^2(\mu)}, are equivalent:

  1. {f} is almost periodic.
  2. The restriction of {T} to {\overline{\{T^nf;n\in\mathbb Z\}}} is a minimal homeomorphism of a compact metric space.
  3. For every {\varepsilon>0}, the set \displaystyle E_\varepsilon=\{n\in\mathbb Z;\left\|T^nf-f\right\|_2<\varepsilon\} is syndetic.

Proof: (a) {\Longrightarrow} (b). Note that the referred restriction is a transitive isometry, by definition. The assertion then follows from the

Exercise 1 Let {(X,d)} be a compact metric space and {T} an isometry on {X}. Then {T} is transitive iff it is minimal.

(b) {\Longrightarrow} (c). It follows from

Exercise 2 Consider a homeomorphism {T} of the metric space {(X,d)}. Then {T} is minimal iff, for any open {U\subset X} and {\varepsilon>0}, the set

\displaystyle \{n\in\mathbb Z;U\cap T^nU\not=\emptyset\}

is syndetic. If in addition {T} is an isometry, the above condition is equivalent to

\displaystyle \{n\in\mathbb Z;d(T^nx,x)<\varepsilon\}

being syndetic for any {x\in X} and {\varepsilon>0}.

(c) {\Longrightarrow} (a). Given {\varepsilon>0}, the syndeticity of {E_\varepsilon} guarantees that the closure of {\{T^nf;n\in\mathbb Z\}} is covered by finitely many balls {B_\varepsilon(T^nf)}. This proves the compacity of {\overline{\{T^nf;n\in\mathbb Z\}}}. \Box

Using (c), we obtain

Proposition 5 Let {\mathbb X=(X,\mathcal B,\mu,T)} be a mps and {f\in L^\infty(\mu)} almost periodic. Then, for any {k\in\mathbb N} and {\varepsilon>0}, the set

\displaystyle E_\varepsilon=\{n\in\mathbb Z;\left\|f\cdot T^nf\cdots T^{kn}f-f^k\right\|_2<\varepsilon\}

is syndetic.

Proof: Suppose {\|T^nf-f\|_2<\varepsilon}. Then {\|T^{jn}f-f\|_2\le k\varepsilon} for every {0\le j\le k}. Hence,

\displaystyle \left|\int f\cdot T^nf\cdots T^{kn}f d\mu-\int f^kd\mu\right|\le k^2\|f\|_\infty^k\varepsilon.

\Box

Corollary 6 (Multiple recurrence for almost periodic functions) Let {f\in L^\infty(\mu)} be a non-negative almost periodic function such that {\int fd\mu>0}. Then, for every {k\ge 1},

\displaystyle \liminf_{N\rightarrow\infty}\frac{1}{N}\sum_{n=1}^N\int f\cdot f\circ T^n\cdots f\circ T^{kn}>0. \ \ \ \ \ (1)

The next theorem will be proved in ERT11.

Theorem 7 A mps {\mathbb X=(X,\mathcal A,\mu,T)} is compact if and only if every {f\in L^2(\mu)} is almost periodic.

The if part is easily obtained by means of simultaneous diophantine approximations. For the converse, we need an algebraic characterization of compact systems, to be established in the next lecture.

Corollary 6 says that compact systems are multiply recurrent in the sense of (1). In particular, considering {f=\chi_A}, we get

Corollary 8 (Multiple Poincaré Recurrence for compact mps) If {\mathbb X=(X,\mathcal A,\mu,T)} is compact and {A\in\mathcal A}, {\mu(A)>0}, then\displaystyle \liminf_{N\rightarrow+\infty}\dfrac{1}{N}\sum_{n=1}^N\mu(A\cap T^{-n}A\cdots\cap T^{-kn}A)>0,

for every {k\ge 1}.

The next post will characterize compact system. They are the Kronecker systems: rotations in abelian groups. This is why the example of {R_\alpha} studied above deserves attention.

Previous posts: ERT0, ERT1, ERT2, ERT3, ERT4, ERT5, ERT6, ERT7, ERT8, ERT9.


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