Hausdorff Dimension, Lagrange and Markov Dynamical Spectra for Geometric Lorenz Attractors

Abstract

In this paper, we show that geometric Lorenz attractors have Hausdorff dimension strictly greater than $2$. We use this result to show that for a “large” set of real functions, the Lagrange and Markov dynamical spectrum associated to these attractors has persistently nonempty interior.

1. Introduction

In 1963 the meteorologist E. Lorenz published in the Journal of Atmospheric Sciences Lor63 an example of a parametrized polynomial system of differential equations,

$$\begin{eqnarray} \dot{x}&=& a(y-x), \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ a=10, \\ \dot{y}&=& rx-y-xz, \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ r=28,{\tag{1}}\\ \dot{z}&=& xy-bz, \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ b={8}/{3}, \end{eqnarray}$$

as a very simplified model for thermal fluid convection, motivated by an attempt to understand the foundations of weather forecasting. Numerical simulations for an open neighborhood of the chosen parameters suggested that almost all points in phase space tend to a strange attractor, called the Lorenz attractor. However Lorenz’s equations proved to be very resistant to rigorous mathematical analysis and also presented very serious difficulties to rigorous numerical study.

A very successful approach was taken by Afraimovich, Bykov, and Shil’nikov ABS77 and by Guckenheimer and Williams GW79 independently: they constructed the so-called geometric Lorenz models for the behavior observed by Lorenz (see section 2 for a precise definition). These models are flows in three-dimensions for which one can rigorously prove the coexistence of an equilibrium point accumulated by regular orbits. Recall that a regular solution is an orbit where the flow does not vanish. Most remarkably, this attractor is robust: it cannot be destroyed by a small perturbation of the original flow. Taking into account that the divergence of the vector field induced by system 1 is negative, it follows that the Lebesgue measure of the Lorenz attractor is zero. Henceforth, it is natural to ask about its Hausdorff dimension. Numerical experiments give that this value is approximately equal to 2.062 (cf. Vis04) and also, for some parameter, the dimension of the physical invariant measure lies in the interval $[1.24063,1.24129]$ (cf. GN16). In this paper we address the problem to prove that the Hausdorff dimension of a geometric Lorenz attractor is strictly greater that $2$. In AP83 and Ste00, this dimension is characterized in terms of the pressure of the system and in terms of the Lyapunov exponents and the entropy with respect to a good invariant measure associated to the geometric model. But, in both cases, the authors prove that the Hausdorff dimension is greater than or equal to $2$, but it is not necessarily strictly greater than $2$. A first attempt to obtain the strict inequality was given in ML08, where the authors achieve this result in the particular case that both branches of the unstable manifold of the equilibrium meet the stable manifold of the equilibrium. But this condition is quite strong and extremely unstable. One of our goals in this paper is to prove the strict inequality for the Hausdorff dimension for any geometric Lorenz attractor; see Figure 1. Thus, our first result is the following.

Theorem A.

The Hausdorff dimension of a geometric Lorenz attractor is strictly greater than $2$.

To achieve this, since it is well known that the geometric Lorenz attractor is the suspension of a skew product map with contracting invariant leaves, defined in a cross-section, we start studying the one-dimensional map $f$ induced in the space of leaves. We are able to prove the existence of an increasing nested sequence of fat (Hausdorff dimension almost $1$) regular Cantor sets of the one-dimensional map (Theorem 1). This fact implies that the maximal invariant set $\Lambda _P$ for the skew product (or, to first return map $P$ associated to the flow) has Hausdorff dimension strictly greater than $1$, and this, in its turn, implies that the Hausdorff dimension of a geometric Lorenz attractor is strictly greater than $2$. In another words, Theorem A is a consequence of the following result.

Theorem 1.

There is an increasing family of regular Cantor sets $C_k$ for $f$ such that

$$\begin{equation*} HD(C_k)\to 1 \ \ \text{as} \ \ k\to +\infty . \end{equation*}$$

The proof of this theorem, although nontrivial, is relatively elementary, and it combines techniques of several subjects of mathematics, such as ergodic theory, combinatorics, and dynamical systems (fractal geometry).

To announce the next goal of this paper, let us recall the classical notions of Lagrange and Markov spectra (see CF89 for further explanation and details).

The Lagrange spectrum $\mathcal{L}$ is a classical subset of the extended real line, related to Diophantine approximation. Given an irrational number $\alpha$, the first important result about upper bounds for Diophantine approximation is Dirichlet’s approximation theorem, stating that for all $\alpha \in \mathbb{R}\setminus \mathbb{Q}$, $|\alpha -\frac{p}{q}|< \frac{1}{q^2}$ has an infinite number of solutions $\frac{p}{q}\in \mathbb{Q}$.

Markov and Hurwitz improved this result by verifying that, for all irrational $\alpha$, the inequality $|\alpha - p/q| < \frac{1}{ \sqrt {5}\cdot q^2 }$ has an infinite number of rational solutions $p/q$, and $\sqrt {5}$ is the best constant that works for all irrational numbers. Indeed, for $\alpha =\frac{1+ \sqrt {5}}{2}$, the gold number, Markov and Hurwitz also proved that, for every $\epsilon > 0,\ |\alpha - \frac{p}{q}|< \frac{1}{(\sqrt {5} + \epsilon ).q^2}$ has a finite number of solutions in $\mathbb{Q}$. Searching for better results for a fixed $\alpha \in \mathbb{R}\setminus \mathbb{Q}$ we are lead to define

$$\begin{equation*} k(\alpha )=\sup \{ k > 0: |\alpha - p/q| < 1/(k\, q^2) \, \text{has infinitely many rational solutions }\, p/q\}. \end{equation*}$$

Note that the results by Markov and Hurwitz imply that $k(\alpha ) \geq \sqrt {5}$ for all $\alpha \in \mathbb{R}\setminus \mathbb{Q}$, and $k(\frac{1+ \sqrt {5}}{2})=\sqrt {5}.$ It can be proved that $k(\alpha ) =\infty$ for almost every $\alpha \in \mathbb{R}\setminus \mathbb{Q}.$

We are interested in $\alpha \in \mathbb{R}\setminus \mathbb{Q}$ such that $k(\alpha )< \infty$ (which forms a set of Hausdorff dimension $1$).

Definition 1.

The Lagrange spectrum $\mathcal{L}$ is the image of the map $k$:

$$\begin{equation*} \mathcal{L}=\{ k(\alpha ), \alpha \in \mathbb{R}\setminus \mathbb{Q}\,\,\text{ and }\,\, k(\alpha )<\infty \}. \end{equation*}$$

In $1921$, Perron gave an alternative expression for the map $k$, as below. Write $\alpha$ in continued fractions: $\alpha =[a_0,a_1,a_2,\ldots ]$. For each $n\in \mathbb{N}$, define

$$\begin{equation*} \alpha _n=[a_n,a_{n+1}, a_{n+2}, \ldots ], \qquad \beta _n=[0, a_{n-1}, a_{n-2}, \ldots ]. \end{equation*}$$ Then

$$\begin{equation} k(\alpha )= \limsup _{n\to \infty }(\alpha _n + \beta _n). {\tag{2}} \end{equation}$$

For a proof of equation 2 see, for instance, CM, Proposition 21.

Markov proved (Mar80) that the initial part of the Lagrange spectrum is discrete: $\mathcal{L}\cap (-\infty , 3)= \{ k_1=\sqrt {5} < k_2=2 \sqrt {2} < k_3=\frac{\sqrt {221}}{5}< \cdots \}$ with $k_n\to 3, k_n^2 \in \mathbb{Q},$ for all $n$.

In $1947$, Hall proved (Hal47) that the regular Cantor set $C(4)$ of the real numbers in $[0,1]$ in whose continued fraction only appear coefficients $1, 2, 3,4$ satisfies $C(4) + C(4) = [\sqrt {2} -1, \, 4 (\sqrt {2} - 1)].$ Using expression 2 and this result by Hall it follows that $[6, \infty ) \subset \mathcal{L}$. That is, the Lagrange spectrum contains a whole half-line, nowadays called a Hall’s ray.

Here we point out $\Lambda = C(4) \times C(4)$ is a horseshoe for a local diffeomorphism related to the Gauss map, which has Hausdorff dimension $HD(\Lambda )>1$. Hall’s result says that its image $f(\Lambda )=C(4) + C(4)$ under the projection $f(x,y)=x+y$ contains an interval. This is a key point to get nonempty interior in $\mathcal{L}$. In $1975$, Freiman proved (Fre75) some difficult results showing that the arithmetic sum of certain (regular) Cantor sets, related to continued fractions, contain intervals, and he used them to determine the precise beginning of Hall’s ray (the biggest half-line contained in $\mathcal{L}$) which is

$$\begin{equation*} \frac{2221564096\, +\, 283748\, \sqrt {462}}{491993569} \cong 4, 52782956616\cdots . \end{equation*}$$

Another interesting set related to Diophantine approximation is the classical Markov spectrum defined by

$$\begin{equation*} \mathcal{M}=\left\{\left(\inf _{(x,y)\in \mathbb{Z}^{2}\setminus (0,0)}|f(x,y)|\right)^{-1}:f(x,y)=ax^2+bxy+cy^2 \ \text{with} \ b^2-4ac=1\right\}. \end{equation*}$$

Notably, the Lagrange and Markov spectrum have a dynamical interpretation. Indeed, the expression of the map $k(\alpha )$ in terms of the continued fraction expression of $\alpha$ given in 2 allows one to characterize the Lagrange and Markov spectrum in terms of a shift map in a proper space. Let $\Sigma =(\mathbb{N}^*)^\mathbb{Z}$ be the set of bi-infinite sequences of integer numbers and consider the shift map $\sigma :\Sigma \to \Sigma ,\ \sigma ((a_n)_n)=(a_{n+1})_n$, and define

$$\begin{equation*} f: \Sigma \to \mathbb{R}, \quad f((a_n)_n)=\alpha _0 + \beta _0, \end{equation*}$$

where $\alpha _0=[a_0,a_{1}, a_{2}, \ldots ]$ and $\beta _0=[0, a_{-1}, a_{-2}, \ldots ]$.

The Lagrange and the Markov spectra are characterized as (cf. CF89 for more details)

$$\begin{equation*} \mathcal{L}=\{\limsup _k f(\sigma ^k((a_n)_n)), \, (a_n)_n \in \Sigma \}, \quad \mathcal{M}=\{\sup _k f(\sigma ^k((a_n)_n)), \, (a_n)_n \in \Sigma \}. \end{equation*}$$

These characterizations lead naturally to a natural extension of these concepts to the context of dynamical systems.

For our purposes, let us consider a more general definition of the Lagrange and Markov spectra. Let $M$ be a smooth manifold, let $T=\mathbb{Z}$ or $\mathbb{R}$, and let $\phi =(\phi ^t)_{t\in T}$ be a discrete-time ($T=\mathbb{Z}$) or continuous-time ($T=\mathbb{R}$) smooth dynamical system on $M$; that is, $\phi ^t:M\to M$ are smooth diffeomorphisms, $\phi ^0=\operatorname {id}$, and $\phi ^t\circ \phi ^s=\phi ^{t+s}$ for all $t,s\in T$.

Given a compact invariant subset $\Lambda \subset M$ and a function $f:M\to \mathbb{R}$, we define the dynamical Markov (resp., Lagrange) spectrum $M(\phi , \Lambda , f)$ (resp., $L(\phi , \Lambda , f)$) as

$$\begin{equation*} M({\phi , \Lambda , f})=\{m_{\phi , f}(x): x\in \Lambda \}, \quad \text{resp.,} \quad L({\phi , \Lambda , f})=\{\ell _{\phi , f}(x): x\in \Lambda \} \end{equation*}$$

where

$$\begin{equation*} m_{\phi , f}(x)\coloneq \sup \limits _{t\in T} f(\phi ^t(x)), \quad \text{resp.,} \quad \ell _{\phi , f}(x)\coloneq \limsup \limits _{t\to +\infty } f(\phi ^t(x)). \end{equation*}$$

It can be proved that $L({\phi , \Lambda , f})\subset M({\phi , \Lambda , f})$ (cf. RM17). In the discrete case, we refer to RM17, where it was proved that for typical hyperbolic dynamics (with Hausdorff dimension greater than $1$), the Lagrange and Markov dynamical spectra have nonempty interior for typical functions.

Moreira and Romaña also proved that Markov and Lagrange dynamical spectra associated to generic Anosov flows (including generic geodesic flows of surfaces of negative curvature) typically have nonempty interior (see RM15 and Rom16 for more details).

Now we are ready to state our next result. Let $X_0$ be the vector field that defines a geometric Lorenz attractor $\Lambda$, and let $U$ be an open neighborhood of $\Lambda$ where $X_0$ is defined.

Theorem B.

Let $\Lambda$ be the geometric Lorenz attractor associated to ${X}_0^{t}$. Then arbitrarily close to ${X^t_0}$, there are a flow ${X}^{t}$ and a neighborhood $\mathcal{W}$ of ${X}^{t}$ such that, if $\Lambda _{Y}$ denotes the geometric Lorenz attractor associated to $Y\in \mathcal{W}$, there is an open and dense set $\mathcal{H}_{Y}\subset C^{1}(U,\mathbb{R})$ such that for all $f\in \mathcal{H}_{Y}$, we have

$$\begin{equation*} \operatorname {int}(L(Y,\Lambda _Y, f))\neq \emptyset , \quad \operatorname {int}(M(Y,\Lambda _Y, f))\neq \emptyset \end{equation*}$$

where $\operatorname {int} \, (A)$ denotes the interior of $A$.

1.1. Organization of the text

This paper is organized as follows. In Section 2, we describe informally the construction of a geometric Lorenz attractor and announce the main proprieties used in the text. In Section 3 we prove the first main result in this paper, Theorem 1 and its consequences, Corollary C and Theorem A. In Section 4 we proof our last result, Theorem B.

2. Preliminary results: geometrical Lorenz model

In this section we present informally the construction of the geometric Lorenz attractor, following GP10AP10, where the interested reader can find a detailed exposition of this construction.

Let $(\dot{x}, \dot{y}, \dot{z})=(\lambda _1 x,\lambda _2 y, \lambda _3 z)$ be a vector field in the cube $[-1,1]^3$, with a singularity at the origin $(0,0,0)$. Suppose the eigenvalues $\lambda _i$, $1\le i\le 3,$ satisfy the relations

$$\begin{equation} 0<-\lambda _3 < \lambda _1<-\lambda _2,\qquad 0<\alpha =-\frac{\lambda _3}{\lambda _1}<1<\beta =-\frac{\lambda _2}{\lambda _1}\,. {\tag{3}} \end{equation}$$

Consider $S\!=\!\{(x,y,1)\!:\! |x|\!\leq \! 1/2, |y|\!\leq \! 1/2\}$ and $S^{-}\!=\!\{(x,y,1) \!\in \! S\!:\! x \!<\! 0\},$ $S^{+}=\{(x,y,1) \in S : x >0\}$, and $S^\star =S\setminus \Gamma$, with $\Gamma =\{(x,y,1)\in S : x=0\}$.

Assume that $S$ is a transverse section to the flow so that every trajectory eventually crosses $S$ in the direction of the negative $z$ axis as in Figure 2. Consider also $\widetilde{\Sigma }^\pm =\{(x,y,z) : x=\pm 1\}$ and put $\Sigma \coloneq \widetilde{\Sigma }^- \cup \widetilde{\Sigma }^+=\{(x,y,z): |x|=1\}$. For each $(x_0,y_0,1) \in S^\star$ the time $\tau$ such that $X^\tau (x_0,y_0,1) \in \Sigma$ is given by $\tau (x_0)= -\frac{1}{\lambda _1}\log (|x_0|)$, which depends on $x_0 \in S^\star$ only and is such that $\tau (x_0) \to + \infty$ when $x_0\to 0$. Hence we get (where $\operatorname {sgn}(x)= \frac{x}{|x|}$ for $x \neq 0)$

$$\begin{equation*} X^\tau (x_0,y_0,1)=(\operatorname {sgn}(x_0), y_0 e^{\lambda _2 \tau (x_0)}, e^{\lambda _3 \tau (x_0)})= (\operatorname {sgn}(x_0), y_0 |x_0|^{-\frac{\lambda _2}{\lambda _1}}, |x_0|^{-\frac{\lambda _3}{\lambda _1}}). \end{equation*}$$

Let $L: S^\star \to \Sigma$ be given by

$$\begin{equation} {L}(x,y,1)=(\operatorname {sgn}(x),yx^\beta ,x^\alpha ). {\tag{4}} \end{equation}$$

It is easy to see that $L(S^\pm )$ has the shape of a triangle without the vertex $(\pm 1,0,0)$, which are cusps points of the boundary of each of these sets. From now on we denote by $\Sigma ^\pm$ the closure of $L(S^\star )$. Note that each line segment $S^\star \cap \{x=x_0\}$ is taken to another line segment $\Sigma \cap \{z=z_0\}$ as sketched in Figure 2. Outside the cube, to imitate the random turns of a regular orbit around the origin and obtain a butterfly shape for our flow, we let the flow return to the cross section $S$ through a flow described by a suitable composition of a rotation $R_\pm$, an expansion $E_{\pm \theta }$, and a translation $T_\pm$. Note that these transformations take line segments $\Sigma ^\pm \cap \{z=z_0\}$ into line segments $S\cap \{x=x_1\}$ as shown in Figure 2, and so does the composition $T_\pm \circ E_{\pm \theta } \circ R_\pm .$ This composition of linear maps describes a vector field $Y$ in a region outside $[-1,1]^3$, such that the time-one map of the associated flow realizes $T_\pm \circ E_{\pm \theta } \circ R_\pm$ as a map $\Sigma ^\pm \to S$. We note that the flow on the attractor we are constructing will pass though the region between $\Sigma ^\pm$ and $S$ in a relatively small time with respect the linearized region.

The above construction enables us to describe, for $t\in \mathbb{R}^+$, the orbit $X^t(x)$ for all $x\in S$: the orbit starts following the linear flow $L$ until $\Sigma ^\pm$ and then it will follow $Y$ coming back to $S$ and so on. Now observe that $\Gamma = \{(x,y,1 \in S : x=0\} \subset W^s((0,0,0))$ and so the orbit of all $x \in \Gamma$ converges to $(0,0,0)$. Let us denote by $W = \{X^{t}(x)\colon x\in S \, ; \, t\in \mathbb{R}^+\}$ the set where this flow acts. The geometric Lorenz flow is the couple $(W, X^{t})$ and the geometric Lorenz attractor is the set

$$\begin{equation} \Lambda =\overline{\bigcap _{t\geq 0}X^{t}(\Lambda _P)}, \,\, \text{where} \,\, \Lambda _{P}=\overline{\bigcap _{i\geq 1}P^{i}(S^\star )}, {\tag{5}} \end{equation}$$ where $P:S^\star \to S$ is the Poincaré map.

Composing the expression in 4 with $R_\pm$, $E_{\pm \theta }$, and $T_\pm$ and taking into account that points in $\Gamma$ are contained in $W^s((0,0,0))$, we can write an explicit formula for the Poincaré map $P$ by

$$\begin{equation*} P(x,y)=(f(x),g(x,y)), \end{equation*}$$

$$\begin{equation*} f(x)=\left\{\begin{array}{l} f_0(x^{\alpha }), \ \ \text{if} \ \ x>0,\\f_1(x^{\alpha }), \ \ \text{if} \ \ x<0, \end{array}\right. \ \ \text{with} \ \ f_i=(-1)^{i}\theta \cdot x +b_i \ \ i=1,2 \end{equation*}$$ and

$$\begin{equation*} g(x,y)=\left\{\begin{array}{l} g_{0}(x^{\alpha },y\cdot x^{\beta }), \ \ \text{if} \ \ x>0,\\g_{1}(x^{\alpha },y\cdot x^{\beta }), \ \ \text{if} \ \ x<0, \end{array}\right. \end{equation*}$$

where $g_1\colon {L_1\times I} \to I$ and $g_{0}\colon {L_2\times I} \to I$ are suitable affine maps, with $L_{1}= [-1/2, 0)$ and $L_{2}=(0,1/2]$. Figure 3 displays the main features of $f$ and $P$ on $[-1/2,\,1/2]$ and $S$, respectively.

2.1. Properties of the one-dimensional map $f$.

Here we specify the properties of the one-dimensional map $f$ described in Figure 3:

($f1$)

$f$ is discontinuous at $x=0$ with lateral limits $f(0^-)=\frac{1}{2}$ and $f(0^+) = -\frac{1}{2}$;

($f2$)

$f$ is differentiable on $I\setminus \{0\}$ and $f'(x) > \sqrt {2}$, where $I=[-1/2,1/2]$;

($f3$)

the lateral limits of $f'$ at $x=0$ are $f'(0^{-}) = +\infty =$ and $f' (0^{+} ) =+\infty$.

The properties $(f1)$–$(f3)$ above imply another important feature for the map $f$, as it is shown in Lemma 2.1 below. We will present the proof of R. Williams (cf. Wil79, Proposition 1) for Lemma 2.1, which we will use to construct “almost locally eventually onto” avoiding the singularity $0$ of $f$ (cf. Section 2.3).

Lemma 2.1.

Put $I=[-\frac{1}{2},\frac{1}{2}]$. If $J\subset I$ is a subinterval, then there is an integer $n$ such that $f^n (J)=I$. That is, f is locally eventually onto.

Proof.

Let $J_0 = J$, if $0\notin J$; otherwise let $J_0$ be the bigger of the two intervals into which $0$ splits $J$. Similarly, for each $i$ such that $J_i$ is defined, set