Decay of Correlations, Quantitative Recurrence and Logarithm Law for Contracting Lorenz Attractors

Abstract

In this paper we prove that a class of skew products maps with non uniformly hyperbolic base has exponential decay of correlations. We apply this to obtain a logarithm law for the hitting time associated to a contracting Lorenz attractor at all the points having a well defined local dimension, and a quantitative recurrence estimation.

Appendix: Linearization and Properties of the Poincaré Map

To study the order of partial derivative of the first return map to the transverse section we use a $C^1$ linearization near the singularity; we will use Theorem 7.1 p. 257 of [15] in its version for flows.

Theorem 8.1

Let $n\in \mathbb {Z}^+$ be given. Then there exists an integer $N = N (n)\ge 2$ such that: if $\Gamma$ is a real non-singular $d\times d$ matrix with eigenvalues $\gamma _1,\ldots ,\gamma _d$ satisfying

$$\begin{aligned} \sum _{i=1}^d m_i\gamma _i\ne \gamma _k \quad \text {for all} \,k=1,\ldots ,d \, \text {and} \, 2\le \sum _{j=1}^d m_i\le N(n) \end{aligned}$$

and if $\dot{\xi }=\Gamma (\xi ) + \Xi (\xi )$ and $\dot{\zeta }= \Gamma \zeta$ , where $\xi ,\zeta \in {\mathbb {R}}^d$ and $\Xi$ is of class $C^N$ for small $||\xi ||$ with $\Xi (0) = 0$, $\partial _\xi \Xi (0) = 0$; then there exists a $C^n$ diffeomorphism R from a neighborhood of $\xi = 0$ to a neighborhood of $\zeta = 0$ such that $R\xi (t)R^{-1}=\zeta (t)$ for all $t \in {\mathbb {R}}$ and initial conditions for which the flows $\zeta (t)$ and $\xi (t)$ are defined in the corresponding neighborhood of the origin.

Since the resonance conditions are open, there exists an $N=N(1)$ such that, if we choose the eigenvalues of the geometric contracting Lorenz Flow respecting the resonance conditions, there exists a $C^N$ neighborhood of $X_0$ such that all the vector fields in the neighborhood respect the resonance conditions and can be $C^1$-linearized. Generically, the linear part of a vector field $\tilde{X}$ in such a neighborhood is different from the linear part of $X_0$; our aim is not to find a common linearization for all the fields in the neighborhood but to ensure the fact that $\tilde{X}$ is $C^1$-linearizable.

1.1 Behaviour Near the Fixed Point

Let $\tilde{X}$ be in a $C^N$ neighborhood of $X_0$ such that the resonance condition are still satisfied; then $\tilde{X}$ can be $C^1$-linearized.

Denote by $\tilde{\lambda }_1,\tilde{\lambda }_2,\tilde{\lambda }_3$ the eigenvalues of $\tilde{X}$ at the fixed point, and denote by $\tilde{r}=-\tilde{\lambda }_2/\tilde{\lambda }_1$ and by $\tilde{s}=-\tilde{\lambda }_3/\tilde{\lambda }_1$.

First, we will study the behaviour of the flow in a neighborhood of the singularity and then use the information about the existence of a foliated atlas to obtain informations on the order of derivatives for the first return Poincaré maps.

Near the singularity, there exists a coordinate system such that the singularity p is in 0, the field is given by $\tilde{X}=(\tilde{\lambda }_1 x,\tilde{\lambda }_2 y, \tilde{\lambda }_3 x)$ and there exists sections $\tilde{\Sigma }=\{z=\varepsilon ,|x|\le \varepsilon ,|y|\le \varepsilon \}$, $\tilde{\Sigma }_+=\{x=+\varepsilon ,\}$ and $\tilde{\Sigma }_-=\{x=-\varepsilon \}$.

By the same computations as for the geometric contracting Lorenz flow $X_0$ we have that the map from $[-\varepsilon ,\varepsilon ]\times [-\varepsilon ,\varepsilon ]\subset \tilde{\Sigma }$ to $\tilde{\Sigma }_+$ is given by:

$$\begin{aligned} \tilde{F}(x,y,1)=(1,\tilde{G}(x,y),\tilde{T}(x))=(\varepsilon ,\varepsilon ^{-\tilde{r}}y\cdot x^{\tilde{r}},\varepsilon ^{1-\tilde{s}} x^{\tilde{s}}), \end{aligned}$$

and that the time taken between $\tilde{\Sigma }$ and $\tilde{\Sigma }_+$ is given by

$$\begin{aligned} \tau (x,y,1)=\frac{\log (\varepsilon )-\log (|x|)}{\tilde{\lambda }_1}. \end{aligned}$$

Remark that both arguments work also for $\tilde{\Sigma }_-$.

We can choose the neighborhood such that $\tilde{s}-1>0$ and $\tilde{r}-\tilde{s}>3$. Therefore, as x approaches 0 we have that

$$\begin{aligned} \frac{\partial \tilde{T}}{\partial x}=O(x^{\tilde{s}-1}),\quad \frac{\partial \tilde{G}}{\partial x}=O(x^{\tilde{r}-1}),\quad \frac{\partial \tilde{G}}{\partial y}=O(x^{\tilde{r}}), \end{aligned}$$

with $\tilde{s}>1$, $\tilde{r}>1$.

1.2 Behaviour Far from the Fixed Point

If $N\ge 3$ we know from [29] that all the vector fields in a neighborhood of $X_0$ preserve a stable foliation. Let $\tilde{X}$ be a vector field in a $C^N$ neighborhood of $X_0$ such that the resonance conditions are preserved, and whose flow preserves a stable foliation. Let $F_{\tilde{X}}(x,y)=(T_{\tilde{X}}(x),G_{\tilde{X}}(x,y))$ be the first return map of the flow to the section $\Sigma$; we will study the behaviour of its partial derivatives as x approaches 0.

With an abuse of notation, we will denote by $\tilde{\Sigma }$, $\tilde{\Sigma }_+$ and $\tilde{\Sigma }_-$ the preimages under the linearizing change of coordinates of the sections $\tilde{\Sigma }$, $\tilde{\Sigma }_+$ and $\tilde{\Sigma }_-$; the important property is that these preimages are locally $C^1$-manifolds and that they are made up of pieces of stable leaves. In particular, we can see $\tilde{\Sigma }_+$ as a graph of a function of (y, z), where the constant leaves are given by constant z.

We want to show that, since the flow preserves the stable foliation, then, recalling the results from Sect. 1 we have that

• (1) the order of $\frac{\partial T}{\partial x}$ as x goes to 0 is the same as the order of $\frac{\partial \tilde{T}}{\partial x}$, i.e., $\tilde{s}-1$

• (2) the order of $\frac{\partial G}{\partial y}$ is the same as the order of $\frac{\partial \tilde{G}}{\partial y}$, i.e., $\tilde{r}$

• (3) the order of $\frac{\partial G}{\partial x}$ is at least the minimum of the orders of $\frac{\partial \tilde{T}}{\partial x}$, $\frac{\partial \tilde{G}}{\partial x}$, $\frac{\partial \tilde{G}}{\partial y}$, i.e., at least $\tilde{s}-1$.

• (4) if $\log (x)$ is integrable with respect to the invariant measure, the return time to $\Sigma$ is integrable.

Since the flow of $\tilde{X}$ has no singularities besides p, the map $\varphi _2(y,z)=(\mu (z),\nu (y,z))$ that takes $\tilde{\Sigma }_+$ into $\Sigma$ is a diffeomorphism that sends lines $z=\text {const}$ into lines $x=\text {const}$ and the map $\varphi _1(x,y)=(\chi (x),\zeta (x,y))$ that takes $\Sigma$ into $\tilde{\Sigma }$ is a map that send lines $x=\text {const}$ into lines $x=\text {const}$.

By direct computation we have that

$$\begin{aligned} DF_{\tilde{X}}&=D\varphi _2\circ D\tilde{F}\circ D\varphi _1\\&=\left[ \begin{array}{cc} \frac{\partial \mu }{\partial z}\frac{\partial \tilde{T}}{\partial x}\frac{\partial \chi }{\partial x} &{} 0\\ \frac{\partial \nu }{\partial y}\bigg (\frac{\partial \tilde{G}}{\partial x}\frac{\partial \chi }{\partial x}+\frac{\partial \tilde{G}}{\partial y}\frac{\partial \zeta }{\partial y}\bigg )+ \frac{\partial \nu }{\partial z}\frac{\partial \tilde{T}}{\partial x}\frac{\partial \chi }{\partial x} &{} \frac{\partial \nu }{\partial y}\frac{\partial \tilde{G}}{\partial y}\frac{\partial \zeta }{\partial y} \end{array} \right] . \end{aligned}$$

Since $\varphi _2$ is a diffeomorphism, therefore $\frac{\partial \nu }{\partial y}\cdot \frac{\partial \mu }{\partial z}\ne 0$, and since $\varphi _1$ is a diffeomorphism $\frac{\partial \chi }{\partial x}\cdot \frac{\partial \zeta }{\partial y}\ne 0$. This proves item (1), (2), (3).

Since the flow has no singularities both the time between $\Sigma$ and $\tilde{\Sigma }$ and the time between $\tilde{\Sigma }_+$ and $\Sigma$ are bounded; this implies item (4).