Statistical properties in hyperbolic dynamics, part 3

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This is the third post of a series of 4 posts based on the lectures at the Houston Summer School on Dynamical Systems 2019 on Statistical properties in hyperbolic dynamics, given by Matthew Nicol, Andrew Török and William Ott. These notes are heavily edited with added comments, examples and explanations. Any mistake is of my responsibility. Some of the notation has been changed for consistency.

Lecture 3

In the previous two lectures we introduced the transfer operator and the martingale approximation to transfer limit laws results from martingales to dynamical systems. In this lecture we will use the decay of correlations of regular observables to deduce statistical properties of dynamical systems.

The setting is as in the previous post: let $T\colon X\to X$ be a non-singular transformation of the probability space $(X,m)$. For observables $\phi,\psi\colon X\to\mathbb{R}$, define their correlation function by

\[C_{\phi,\psi}(n) = \int_X \phi\cdot \psi\circ T^n dm - \int_X \phi dm \int_X\psi dm.\]

Note that $C(\phi+a,\psi +b) = C(\phi ,\psi)$ for any pair of real constants $a,b$, so we can assume that the observables have zero mean. The first question we are interested in solving is:

Question: can we find an invariant measure with good properties?

By good properties we mean a measure not supported on periodic orbits and absolutely continuous with respect to the reference measure $m$.

Perron-Frobenius theorem

We recall the well celebrated theorem in one if its probabilistic formulations:

Theorem (Perron-Frobenius):

Let $A$ be a positive matrix. Then there exists a maximal simple real eigenvalue $\lambda$, and its associated eigenvector can be chosen with positive entries. The rest of the eigenvalues are contained in a disk of radius strictly smaller than $\lambda$.

The previous theorem describes de structure of the spectrum of a positive matrix. One of the main consequences of this theorem is the construction of a stationary measure for Markov chains. We will not go into the details of the proof of the theorem as that will be studied later on in the context of Birkhoff cones. In what follows we will focus on the infinite dimensional analogue of the Perron-Frobenius theorem.

Infinite dimensional results

The previous theorem can be extended for infinite dimensional spaces. The notions of eigenvalues and spectrum are more delicate in the infinite dimensional case, so we will define the right notions to generalize the previous theorem.

From now on, let $(\mathcal{B},| \cdot |)$ a Banach space and $L\colon \mathcal{B}\to \mathcal{B}$ a bounded linear operator with spectral radius $\rho(L)$. We say that $L$ has spectral gap if it has a single simple eigenvalue on the circle {$ z : |z | = \rho(L) $} and the rest of the spectrum is contained in a disk of radius strictly smaller than $\rho$.

While this is the property we want to obtain for the transfer operator, we will define a weaker notion which is easier to prove.

We say that $L$ is quasi-compact if $\mathcal{B}$ can be written as direct sum $\mathcal{B} = F\oplus H$ and a positive number $0 < \rho < \rho(L)$ such that

  1. $F$ and $H$ are closed and $L$ invariant,
  2. $\dim(F) < \infty$ and all the eigenvalues of $L|_F$ have modulus larger than $\rho$,
  3. The spectral radius of $L|_H$ is strictly smaller than $\rho$.

Quasi-compactness and spectral gap holds trivially for operators in finite dimensional spaces. If an operator is quasi-compact, there is a unique eigenvalue on the circle {$z : |z | = \rho(L) $} and this is simple, then $L$ has spectral gap. In particular, the decomposition of $\mathcal{B}$ as $F\oplus H$ has the additional property that $\dim(F) = 1$.

There are multiple theorems that give conditions for the transfer operator to be quasi-compact or have spectral gap, by Lasota & Yorke, Doeblin & Fortet, Ionescu–Tulcea & Marinescu, among others. We will state a theorem by Hennion which applies in great generality.

Theorem (Hennion): suppose there is a semi-norm $|\cdot |’$ on $\mathcal{B}$ such that:

  1. $|\cdot|’$ is continuous with respect to $|\cdot|$,
  2. The unit ball of $\mathcal{B}$ is compact with respect to the semi-norm $|\cdot|’$,
  3. $L$ is bounded with respect to $|\cdot|’$,
  4. There are constants $k\geq 1$, $r\in(0,\rho(L))$ and $R > 0$ such that
\[| L^k v | \leq r^k|v| + R| v|'\]

for all $v\in \mathcal{B}$. Then $L$ is quasi-compact.

Property 4 is often called Doeblin-Fortet inequality.

Consequences of spectral gap

Absolutely continuous invariant measure: Suppose that the transfer operator $P:\mathcal{B}\to\mathcal{B}$ has spectral gap for some Banach space of functions $\mathcal{B}\subset L^1$ with respect to the semi-norm $|\cdot|’$. Recall that the spectral radius of $P$ is bounded above by $1$ in $L^1$, and so is in $\mathcal{B}$. Write the decomposition of the Banach space $\mathcal{B} = F\oplus H$ as above. Since $\dim(F) = 1$, we can pick a basis of $F$ consisting of an eigenfunction of $P$, that is, there is $f\in F$ and $c_F\in\mathbb{C}$ such that $P f = c_F f$. For simplicity of the arguments we will assume $c_F = 1$, which is often the case with applications (otherwise normalize). We can take $f$ normalized so that $\int_X f dm = 1$. Define the measure $\mu$ by

\[\mu(A) = \int_X f dm.\]

The condition $\int_X f dm = 1$ ensures that $\mu$ is a probability measure, and it is trivially absolutely continuous with respect to $m$. We see now that $\mu$ is $T$-invariant:

\[\int_X \phi \circ T d\mu = \int_X \phi \circ T \cdot f dm = \int_X \phi \cdot P(f) dm = \int_X \phi f dm = \int_X \phi d\mu\]

which shows that $\mu$ is invariant.

Exercise: prove that the simplicity of the eigenvalue implies that $\mu$ is ergodic.

Decay of correlations: Let $\psi\in L^\infty$ and $\phi\in\mathcal{B}$. We will suppose that $\mathcal{B}$ is closed under products and $\mathcal{B}\subset L^\infty$. The correlation function with respect to $\mu$ for observables $\phi,\psi\in\mathcal{B}$ is given by:

\[C_{\phi,\psi}^\mu(n) = \int_X \phi\cdot \psi\circ T^n fdm - \int_X \phi fdm \int_X\psi fdm.\]

Define the operator $\Pi\colon L^1\to F$ by $\Pi(\phi) = m(\phi) f$, and the complementary operator $N\colon L^1\to H$ by $N(\phi) = P(\phi) - \Pi(\phi)$. Note that $\Pi$ and $N$ commute, so $P^n = \Pi^n + N^n$. Then we can write the correlation function as

\[C_{\phi,\psi}^\mu(n) = \int_X P^n(\phi f)\psi dm - \int_X \Pi^n(\phi f)\psi dm = \int_X N^n(\phi f) \psi dm.\]

Using the continuity of the semi-norm, the fact that $P|_H = N$ and that $\psi\in L^\infty$, we obtain

\[|C_{\phi,\psi}^\mu(n)| \leq |N^n(\phi f) \psi |_{L^1(m)} \leq |\psi|_{L^\infty}|N^n(\phi f)|_{L^1(m)} \leq |\psi|_{L^\infty}|N^n(\phi f)|_{\mathcal{B}}\] \[\leq |\psi|_{L^\infty}|P^n(\phi f)|' \leq |\psi|_{L^\infty} (\rho + \epsilon)^n\]

where $\epsilon > 0$ is small enough so that $(\rho + \epsilon) < 1$. This proves that for regular enough observables, we have exponential decay of correlations.

Explicit example

One of the main sources of spectral gap is uniform hyperbolicity. We will show with a simple example how the theory discussed above can be applied in a situation of uniform hyperbolicity. For simplicity, our map will be measure preserving (the reference measure), which makes the computations slightly more simple to write.

Consider the map $T\colon [0,1]\to [0,1]$ given by $T(x) = 2x \pmod 1$ and the reference measure $m = \operatorname{Leb}$. Note that $T$ is $m$-invariant. Let $P$ be the corresponding transfer operator, acting on $L^1(m)$. Let $\mathcal{B}$ be the space of Lipschitz functions on $[0,1]$, with the semi-norm

\[|\phi|_{Lip} = \sup_{x\neq y}\dfrac{|\phi(x)-\phi(y)|}{|x-y|}.\]

Note that this semi-norm vanishes for all constant functions. We can either mod out $\mathcal{B}$, or modify the definition of the norm, adding a contributuion of the supremum norm. We chose the later, and define $\| \cdot\|_{Lip}$ as

\[\|\phi\|_{Lip} = |\phi|_{Lip} + |\phi|_{L^1}.\]

Note that the constant function $\mathbf{1}$ is a fixed point of $P$ (or an eigenfunction with eigenvalue equal to $1$). Define the subspace $F$ of $\mathcal{B}$ as the generated by the function $\mathbf{1}$. We define $H$ as the subspace of functions in $\mathcal{B}$ with zero integral (with respect to the reference measure $m$). It is easy to see that $\mathcal{B} = F\oplus H$, as we can write every function $\phi\in\mathcal{B}$ as $\phi = m(\phi)\mathbf{1} + (\phi - m(\phi)\mathbf{1})$. We now examine the spectrum of $P$ restricted to $H$.

Proposition: $|P\phi |_{Lip}\leq \frac{1}{2} | \phi|_{Lip}$ for all $\phi\in\mathcal{B}$.

Proof: see last example of lecture 1.

With this, the fact that $m(P\phi) = m(\phi)$ and the Gelfand formula for the spectral radius we can see that

\[\rho(P|_{H}) \leq \| P|_H \|_{Lip} = \sup_{\phi\in H\setminus\{ 0\}} \dfrac{\| P\phi \|_{Lip}}{\| \phi \|_{Lip}} = \sup_{\phi\in H\setminus\{ 0\}} \dfrac{|P\phi|_{Lip}}{|\phi|_{Lip}} \leq \dfrac{1}{2}.\]

This proves that our systems has spectral gap.

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