A probabilistic approach to intermittency

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In this entry, we will discuss some of Carlangerlo Liverani’s work. Without any doubt, his work Decay of Correlations, Annals of Mathematics, 142, pp. 239-301, (1995) is one of the most influential of his publications, but this time we will focus on his work with Sandro Vaienti and Benoit Saussol, A Probabilistic Approach to Intermittency, Ergodic Theory and Dynamical Systems, 19, pp. 671–685 (1999). We will refer to this paper as LSV99.

In previous entries (see here, here) we have seen how uniform hyperbolicity yields good spectral properties for the transfer operator associated to a dynamical system. Under general assumptions, the transfer operator acts on a suitable Banach space of functions with spectral gap, that is, the spectrum of the operator consists of a maximal real simple eigenvalue, and the rest of the spectrum is contained in a disk of radius strictly smaller than the length of the maximal eigenvalue. This has several dynamical consequences: existence of absolutely continuous invariant probability measures, exponential mixing (and hence ergodicity) of such measures, decay of correlations, among others properties. For the case of intermittent maps (or in general, non-uniformly hyperbolic maps), the transfer operator does not posses such properties, that is, there is an absence of spectral gap. Thus, the spectral techniques are not available.

We start by defining the class of maps that we will study: the Liverani-Saussol-Vaienti (LSV) maps are defined by $T_\alpha(x) = x(1+2^\alpha x^\alpha)$ if $x\in [0,1/2)$ and $T_\alpha(x) = 2x - 1 $ for $x\in [1/2,1]$, for $\alpha\in (0,1)$. All of the maps of this family have an indifferent fixed point, that is, $(T^n_\alpha)’(0) = 0$ for all $\alpha$ and $n\geq 1$. We will denote by $m$ the Lebesgue measure on $X = [0,1]$. If we identify the points $0$ and $1$ it is clear that the maps $T_\alpha$ become continuous.

It is easy to see that this measure is not invariant under any of these transformations. One of the insights of LSV99 is using the theory of invariant cones to address several of the ergodic questions. First we recall the definition of the transfer operator: for a fixed transformation $T_\alpha$, its transfer operator $P\colon L^1(m)\to L^1(m)$ is defined as the dual of the composition operator $U\colon L^\infty(m)\to L^\infty(m)$ given by $Uf(x) = f\circ T_\alpha(x)$ for $f\in L^\infty(m)$. The duality relation is given by

\[\int_X Pf g dm = \int_X f Ug dm\]

for all $f\in L^1(m)$ and $L^\infty(m)$. The transfer operator can also be understood as the operator giving the density of the push-forward measure of an absolutely continuous measure under the dynamics, that is, if $d\mu_f = f dm$ then $d(T_*(\mu_f)) = P(f)dm$ where $T_*(\nu)(A) = \nu(T^{-1}(A))$. Note that from this it follows that a measure $\mu_f$ is $T$ invariant if and only if $Pf = f$. Finally, an explicit expression for the transfer operator (with respect to the reference measure $m$) can be given:

\[Pf(x) = \sum_{Ty = x} \dfrac{f(y)}{T'(y)}.\]

This expression allows us to perform concrete calculations and estimates. One of the main features of LSV99 is the use of invariant cones (this has been done before for instance in Decay of Correlations, Annals of Mathematics, 142, pp. 239-301, (1995) by Liverani).

Recall the definition of a cone: a subset $\mathcal{C}$ of a vector space $V$ is a cone if for any $f,g\in\mathcal{C}$ and $a,b\in [0,\infty)$ one has $af+bg \in \mathcal{C}$. We define now the cones: let $\mathcal{C}_0 = \{ f \in C^0((0,1])) \ | \ f \geq 0 , f \text{ is decreasing} \}$. The transfer operator preserves $\mathcal{C}_0$ as it is a positive operator. Let now $X(x) = x$ be the identity function and $m(f)$ be the integral of $f$ with respect to $m$, and defined the cone $\mathcal{C}_2$ by

\[\mathcal{C}_2 = \{ f\in\mathcal{C}_0 \cap L^1([0,1]): X^{\alpha + 1}f \text{ is increasing}, f(x)\leq ax^{-\alpha}m(f) \}.\]

If $a\geq 2^\alpha(\alpha+2)$, then the transfer operator preserves this cone. Proving this need a more involved analysis of the transfer operator associated to the LSV maps, using its explicit form mentioned above. The idea here is to think of functions in $\mathcal{C}_2$ as functions which behave essentially like $x^{-\alpha}$, that is, they are well behaved in $(0,1)$ but they may have a singularity around $0$ which looks like $x^{-\alpha}$. The condition $X^\alpha f$ being increasing means that this singularity can be fixed by multiplying by $x^\alpha$. With the invariance of this cone, we can already prove the existence of an absolutely continuous invariant probability measure:

Theorem: there exists a locally Lipschitz function $h$ such that $Ph = h$ and $h(x)\leq ax^{-\alpha}$.

Proof: consider the set of functions $K = \{ f\in\mathcal{C}_2 : m(f) = 1 \}$ and $H = X^{\alpha + 1}K$. The functions in $H$ functions are equibounded and equicontinuous: note that if $f\in K$ then $0\leq X^{\alpha+1}f$ is increasing and continuous in $(0,1]$, hence it reaches its maximum. On the other hand, for $x\geq y > 0$ we have

\[0 \leq x^{\alpha+1}f(x) - y^{\alpha+1}f(y) \leq (x^{\alpha+1} - y^{\alpha+1})f(x) \leq (x^{\alpha+1} - y^{\alpha+1})ax^{-\alpha}\]


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