Conditional mixing in deterministic chaos

Abstract

While on the one hand, chaotic dynamical systems can be predicted for all time given exact knowledge of an initial state, they are also in many cases rapidly mixing, meaning that smooth probabilistic information (quantified by measures) on the system’s state has negligible value for predicting the long-term future. However, an understanding of the long-term predictive value of intermediate kinds of probabilistic information is necessary in various physical problems, and largely remains lacking. Of particular interest in data assimilation and linear response theory are the conditional measures of the Sinai–Ruelle–Bowen (SRB) measure on zero sets of general smooth functions of the phase space. In this paper we give rigorous and numerical evidence that such measures generically converge back under the dynamics to the full SRB measures, exponentially quickly. We call this property conditional mixing. While conditional mixing typically cannot be proven from standard transfer operator theory, we will prove that conditional mixing holds in a class of generalized baker’s maps, and demonstrate it numerically in some non-Markovian piecewise hyperbolic maps. Conditional mixing provides a natural limit on the effectiveness of long-term forecasting of chaotic systems via partial observations, and appears key to proving the existence of linear response outside the setting of smooth uniform hyperbolicity.

1 Introduction

By definition, chaotic dynamical systems’ future states are generally impossible to predict over the long term. For this reason they must be studied probabilistically, that is, in terms of measures that evolve under the dynamics. Many probabilistic questions about topologically mixing chaotic systems $f: M \circlearrowleft $ are quantitatively related to fast mixing (that is, decay of correlations) for smooth observables with respect to some invariant measure $\mu $ [2]. Mathematically, mixing means the following decay of sufficiently regular (for example, $C^\infty $ ) observables $A, B: M \to \mathbb {R}$ :

$$ \begin{align*} \int_M A \circ f^n \, B\, \mathrm{d}\mu - \int_M A\,\mathrm{d}\mu \int_M B\,\mathrm{d}\mu \xrightarrow{n\to\infty} 0. \end{align*} $$

Alternatively, this can be reformulated as weak convergence of smoothly reweighted versions of $\mu $ back to $\mu $ under the actions the dynamics:

$$ \begin{align*} \int_M A \, \mathrm{d} f^n_* (B \mu) \xrightarrow{n\to\infty} \int_M A\,\mathrm{d}\mu \end{align*} $$

for any $A,B$ with $\int _M B\,\mathrm {d}\mu = 1$ , where $f_*$ pushes measures forward under f.

A particularly important invariant measure is the Sinai–Ruelle–Bowen (SRB) measure $\rho $ . This is typically the most physically relevant invariant measure: among other things, assuming no periodicity in the system, measures with smooth enough Lebesgue densities commonly converge to the physical measure over time (typically exponentially quickly), that is,

$$ \begin{align*} { \int_M A \, \mathrm{d} f^n_* (B \operatorname{\mathrm{Leb}}) \xrightarrow{n\to\infty} \int_M A\,\mathrm{d}\rho }\end{align*} $$

for all $A,B \in C^\infty $ with $\int _M B\,\mathrm {d}\operatorname {\mathrm {Leb}} = 1$ . It is a standard meta-result in smooth ergodic theory that for sufficiently hyperbolic maps with $d_u$ positive Lyapunov exponents, the same result also holds for any measure $\mu $ with a regular conditional density along $d_u$ -dimensional submanifolds that are tangent to the expanding direction in phase space:

(1) $$ \begin{align} \int_M A \, \mathrm{d} f^n_* (B \operatorname{\mathrm{Leb}}) \xrightarrow{n\to\infty} \int_M A\,\mathrm{d}\rho \end{align} $$

However, establishing fast mixing (or the related decay of iterates of a transfer operator) appears to be insufficient to answer various questions that depend on the long-term behaviour of a small problematic subset of the system’s attractor, such as the response problem in non-uniformly hyperbolic systems [21]. We might therefore ask if other measures $\mu $ also satisfy (1).

Perhaps the simplest example of a small subset of an attractor would be its restriction to a submanifold of phase space—which will generically be transverse to both stable and unstable manifolds. If this submanifold comes from some foliation (for example, of level sets of an observable), we can disintegrate the physical measure and study its conditional measure $\mu $ on this submanifold. An example of such a measure is shown in Figure 1: note that unlike Gibbs invariant measures, the measure is supported on a non-invariant Cantor set without a product structure.

Figure 1 Left: picture of the Lozi attractor at $a=1.7, b=0.5$ (black), the singular line $\{x = 0\}$ (purple line), their intersection (pink Cantor set). Right: histogram of $\rho (y\mid 0)$ , obtained from $200,\!000$ iterates of the unstable manifold dynamics $\vec {f}$ , binned at width $0.0025$ .

If (1) obtains for such a conditional measure $\mu $ , we call it conditional mixing. For dissipative systems, conditional mixing appears to lie outside the scope of traditional study by transfer operators, and this basic problem has hitherto seen very little study, notwithstanding work for other classes of $\mu $ in the specific case of linear one-dimensional maps [7, 15, 22]

Nevertheless, through a mixture of rigorous and theoretical study we will show that conditional mixing holds for a range of maps, and, in particular, has some connection with the fractal-geometric theory of Fourier dimension [19, 22]. From our results it seems that conditional mixing is likely to hold for a large set of maps and submanifolds (perhaps even almost all those for which there is no ‘obvious’ reason why it could not hold).

We also describe some consequences of conditional mixing. One physically meaningful consequence of conditional mixing is that the capability of Bayesian filters to make predictions about chaotic systems in the long term is limited if they only make partial observations (see §3); it also implies set-filling results for chaotic attractors (see §2). Another physical application, to proving the widely believed existence of linear response outside of smooth hyperbolic systems, is considered in [24].

We stress that, except for conservative systems, conditional mixing does not follow from traditional Banach space approaches [2, 3]: these typically rely on some sort of regularity of the initial measure (for example, smoothness of a density along a foliation). The conditional measures we will study in this paper have no regularity of this sort, and indeed are Cantor measures supported on totally disconnected sets. Banach space approaches, furthermore, typically proceed by showing that this kind of regularity is preserved or improved, and results usually hold while the map’s structure remains the same: on the other hand, we present an example in §4.1 where one choice of map fails to have conditional mixing even though all the structurally identical maps around it do. We note that the Fourier dimension theory for nonlinear maps [7, 22] uses the Dolgopyat method (in dynamics more commonly used to prove rapid mixing for flows) [11], where one is interested in joint non-integrability.

The paper is structured as follows. In §2 we give a mathematical definition of conditional mixing and give a simple, illustrative consequence of it involving set filling. In §3 we present the application of conditional mixing to the area of forecasting. In §§4 and 5 we give evidence for conditional mixing in various systems, respectively presenting a theorem for a class of (potentially nonlinear) baker’s maps and numerical evidence for some piecewise hyperbolic maps. We discuss our results in §6. The novel algorithms we use to obtain for our numerical results are given in Appendix A.

2 Definition and an illustrative consequence

Let $T: M\to M$ be a dynamical system with SRB measure $\rho $ , and let $H: M \to \mathbb {R}^d$ be a $C^2$ function with no critical points on the level set

$$ \begin{align*}\ell_H := \{x \in M : H(x) = 0\}.\end{align*} $$

Suppose that for some $c:\mathbb {R}^+ \to \mathbb {R}^+$ the following limit exists in the $C^0$ -weak topology:

Suppose also that $\mu $ is a finite measure. Any two sequences $c, c'$ will yield $\mu , \mu '$ identical up to scaling, as would changing the kernel

to another bounded, compactly supported decreasing function. If $\mu $ is a probability measure then we denote it by $\mathrm {d}\mu (x) = \mathrm {d}\rho (x \mid H(x) = 0)$ .

Definition 2.1. $(T,\rho )$ has conditional mixing with respect to H if for all $A, B \in C^\infty $ ,

(2) $$ \begin{align} \bigg|\int_M A\circ T^n\,B\, \mathrm{d}\mu - \int_M A\,\mathrm{d}\rho \int_M B\, \mathrm{d}\mu\bigg| \xrightarrow{n\to\infty} 0. \end{align} $$

This is to say that repeatedly pushing forward any weighted conditional measure $B\mu $ by T, we converge back to the SRB measure $\rho $ (up to a multiplicative constant) in the weak topology with respect to $C^1$ .

It is natural to wish to put some quantitative bounds on mixing. By analogy with the usual sort of mixing, a natural decay rate is exponential.

Definition 2.2. $(T,\rho )$ has exponential conditional mixing with respect to H if there exist $\xi <1$ , C and $r<\infty $ such that for all $A, B \in C^r$ ,

$$ \begin{align*} \bigg|\int_M A\circ T^n\,B\, \mathrm{d}\mu - \int_M A\,\mathrm{d}\rho \int_M B\, \mathrm{d}\mu\bigg| \leq C \xi^n \|A\|_{C^r} \|B\|_{C^r}. \end{align*} $$

Note that (exponential) conditional mixing can already be expected to hold from the classical theory if T is conservative. For example, we have the following proposition (proven in Appendix D).

Proposition 2.1. Suppose $T: M \to M$ is a $C^3$ conservative topologically mixing Anosov diffeomorphism with M a compact $C^\infty $ manifold. Suppose $H \in C^2(M,\mathbb {R}^d)$ has no singular points on its zero set $\ell _H$ , the zero set is everywhere transverse to stable manifolds, and d is less than or equal to the number of stable directions. Then $(T,{\operatorname {\mathrm {Leb}}})$ has exponential conditional mixing with respect to H.

The measure-based conditional mixing implies an interesting set-convergence property of the intersection of the level set $\ell _H$ with the support of $\rho $ , which we denote by $\Lambda $ and which is often an attractor of T. We find that iterates of the intersection of the line and $\Lambda $ , that is, iterates of a slice of $\Lambda $ , converge back in Hausdorff distance to the full support of $\rho $ . An example of this phenomenon is plotted in Figure 2.

Figure 2 Exponentially fast filling of the Lozi attractor $\Lambda \subset \mathbb {R}^2$ (black) by the set intersection $\Lambda \cap \{x = 0\}$ (pink), pushed forward n times by the same Lozi map. The level set $\{x = 0\}$ is given in purple. Lozi parameters $a = 1.8$ , $b = 0.35$ are used.

Proposition 2.2. Conditional mixing with respect to H implies that

$$ \begin{align*} \lim_{n\to\infty} d_{\mathrm{Haus}}(T^n(\ell_H \cap \Lambda),\Lambda) = 0. \end{align*} $$

Under a reasonably general assumption on the regularity of the SRB measure, exponential conditional mixing gives us a quantitative version of this result as well.

Proposition 2.3. Suppose $\rho $ is lower-Ahlfors regular: that is, there exist $C,d$ such that $\rho (B(x,\delta ))> C \delta ^d$ for all x. Exponential conditional mixing with respect to H implies that for some $\xi _1 < 1$ and $C_1$ ,

$$ \begin{align*} d_{\mathrm{Haus}}(T^n(\ell_H \cap \Lambda),\Lambda) \leq C_1\xi_1^n. \end{align*} $$

The proofs of these two propositions are given in Appendix D.

3 Forecasting with perfect partial observations

We now consider a fundamental practical problem to which the notion of conditional mixing is directly applicable: that of forecasting chaotic dynamics. Many forecasting methods have been developed that assimilate information obtained from observations. In general, these methods achieve this assimilation by approximating the Bayesian filter, also known as the optimal filter [12]. The stability of the Bayesian filter with small, finite observation error has recently been studied in Anosov systems [20].

To understand this in the simplest instance, let us suppose that our system $T: \mathcal {M} \circlearrowleft $ for $\mathcal {M} \subset \mathbb {R}^d$ has exponential mixing, and at time $n=0$ we have some prior probabilistic knowledge of the state of our system, given by some (presumably ‘nice’) measure $\mathrm {d}\mu ^{-}(x)$ . If we start with an unobserved system at statistical equilibrium, the natural choice of prior is $\mu ^{-} = \rho $ , the SRB measure.

We can now make a noisy and perhaps partial observation of our system, given as a value $y = H(x) + \zeta \in \mathbb {R}^{e}$ , where $\zeta $ is random with probabilities given by a smooth kernel $p(\zeta \mid x)\,\mathrm {d} \zeta $ .

Assimilating the observation y, the posterior probability measure of x is given by Bayes’ theorem as

(3) $$ \begin{align} \mathrm{d}\mu(x) = Z(y)^{-1} p(y - H(x)\mid x)\,\mathrm{d}\mu^{-}(x), \end{align} $$

with normalizing constant $Z(y) = \int _{M} p(y-H(w)\mid w)\,\mathrm {d}\mu ^{-}(w)$ . An example is given in Figure 3. Our best guess of the state of the system at future time n (that is, of $T^n(x)$ ) is then given by $T^n_*\mu $ ; the expected value of some (nice) observable A at time n is therefore

(4) $$ \begin{align} \int_M A(T^n(w))\,\mathrm{d}\mu(w) = \int_M A(T^n(w))\, \frac{p(y-H(w)\mid w)}{Z(y)}\,\mathrm{d}\mu^-(w). \end{align} $$

Figure 3 Posterior distribution (pink) after one observation step of $H(u,v) = u + \zeta $ where $\zeta \sim \mathcal {N}(0,\sigma ^2)$ , for varying values of $\sigma $ . Line corresponding to $H(x,y) = H_{\mathrm {obs}}$ in purple, attractor in black. Convergence to the conditional measure as in Figure 1 can be seen as the noise $\sigma \to 0$ .

Depending on how effective a measurement $y=H(x)$ is of x, $\mu $ is likely to be concentrated on a smaller set than $\mu ^-$ , and this may improve forward estimates of the system’s state over the short to medium term. However, under our assumptions for $T, A, \mu ^-, p$ , exponential mixing results give that (4) will eventually converge at a fixed exponential rate to the SRB measure expectation $\int _M A\,\mathrm {d}\rho $ .

But as we make our observations more and more precise, that is, reduce the noise in H to zero, we might ask what posterior we end up with and what quality of forecast we can make with it. This is trivial if $H(x)$ specifies x: we will know our value of x exactly, and therefore $T^n(x)$ exactly for all time. However, it is typical in high-dimensional systems for H to be only a partial observation.

In a zero-noise limit the kernel p is no longer smooth, with $p(\zeta \mid w) = \delta (\zeta )$ . Our posterior measure $\mathrm {d}\mu (x)$ is then the simply the conditional probability measure of $\mathrm {d}\mu ^{-}(x)$ given that $H(x) = y$ :

$$ \begin{align*}\int_M A(f^n(w))\,\mathrm{d}\rho(w\mid H(w) = y). \end{align*} $$

If conditional mixing (2) holds, this must converge to the expectation of A with respect to the SRB measure $\rho $ , that is, in the long term we end up back with the default no-information guess. Conditional mixing thus codifies the intuition that typical incomplete information on the system should wash out over time.

On the other hand, if conditional mixing did not hold generically, and thus partial observations could with some positive likelihood be predictively useful for all time, there might be significant practical consequences for chaotic systems. However, what our exploratory mathematical results in §§4 and 5 will suggest is that conditional mixing should in fact hold, excluding this possibility.

In practice, of course, physical observations will always have some random error to them. This is also true of the evolution of physical systems, which are nonetheless considered worth studying in their zero-noise limit. In any case, if the observation uncertainty is small enough it will take some time to manifest in the prediction, and the behaviour until then is described by zero-noise limit we have discussed here (see Figure 4).

Figure 4 Decay of mean forecasts of $A(x,y) = 2x$ to SRB measure averages for Bayesian filters with various levels of observation noise. The observation function used was $H(x,y) = x$ .

Nevertheless, prediction over the long term with small, non-zero observation errors does admit an interesting question for which general results can be proven using standard dynamics techniques. If T is a $C^\infty $ Anosov diffeomorphism and $A \in C^\infty (M)$ , then over the long term, the convergence of the forward predictions (4) is governed by how much $\mu $ picks up the transfer operator eigenfunctions of T. Mathematically, we have for all $K\in \mathbb {N}$ that as $n\to \infty $ [14],

(5)

where $r_k\in \mathbb {N}$ and $1> |\unicode{x3bb} _1| \geq |\unicode{x3bb} _2| \geq \cdots \to 0$ . Both the $\alpha _k$ and $\beta _k$ are independent of the Bayesian posterior $\mu $ : the $\alpha _k,\beta _k$ are functionals, and in particularly $\beta _k$ is an integration against a hyperdistribution, that is, $C^\infty $ in the unstable direction. While the problem is technical and beyond the scope of this paper, this latter fact suggests that as the observation error decreases to zero (and thus the density ${\mathrm {d}\mu }/{\mathrm {d}\rho }$ converges to a conditional distribution supported on $\ell _H$ ), the integral $\beta _k({\mathrm {d}\mu }/{\mathrm {d}\rho })$ could potentially converge to a finite limit when $\ell _H$ is transverse to stable manifolds, at least for Anosov maps with a $C^\infty $ unstable foliation. If this were the case, it would suggest that reducing the observation error does not strengthen the asymptotic long-term predictive power of the Bayesian filter—that is after any transients, which will at least initially approximate the zero-noise limit, and may take arbitrarily long to dissipate.

However, $C^\infty $ uniformly hyperbolic systems are probably unique in that their transfer operators have no non-removable essential spectrum, and K can therefore be taken to infinity in (5). We can thus contrast the $C^\infty $ Anosov case to our numerical evidence for the Lozi map in Figure 4, where the predictive power appeared to be improved by smaller error at times when the conditional mixing regime had washed out. Note that the Lozi map has a rough unstable foliation [4] and can usually be expected to lack a finite Markov partition [17], leading to essential spectrum for its transfer operator [8].

This argument would not yield a proof of conditional mixing a priori, as bounds for the error in the asymptotic expansion (5) will blow up as the observation noise goes to zero.

4 Rigorous results for a toy model: baker’s map

As a simple model to study conditional mixing, let us consider baker’s maps $b: D := [0,1]^2 \circlearrowleft $ of the following form:

(6) $$ \begin{align} b(x,y) = (kx \,\mod\! 1, v_{\lceil kx \rceil}(y)), \end{align} $$

where $k\geq 2$ is an integer, the $v_i, i = 1,\ldots ,k$ , are (possibly nonlinear) contractions with all $|v_i'|\leq \mu < 1$ . We will also assume that the open images $v_i((0,1))$ are disjoint, and the contractions have bounded distortion (that is, the $\log |v_i'|$ are $C^1$ ). An example of such a map is plotted in Figure 5. For smooth enough transverse foliations, it is possible to define in a natural way conditional measures for all leaves. For simplicity, we assume we have a foliation $\Psi (t;y)$ of a subset of D into graphs $x = \psi (y) - t$ , with the parameter t translating the graph in the x direction.

Proposition 4.1. Suppose that $\Psi (t;y) = (\psi (y) -t,y)$ is a foliation of some subset $D_\Psi \subseteq D$ for $|t| \leq t_* \in [0,1]$ and $y \in [0,1]$ . Let $\rho $ be the (unique) SRB measure of b.

Then for every $t\in [-t_*,t_*]$ there exists a unique probability measure $\rho _t$ supported on $\Psi (t,[0,1])$ such that the following assertions hold.

1. (a) For all $t \in (-t_*,t_*)$ and all continuous functions $A: D\to \mathbb {R}$ ,

   (7) $$ \begin{align} \int_D A\,\mathrm{d}\rho_t = \lim_{\delta\to 0} \frac{1}{2\delta} \int_{\{\Psi(s,y) : |s-t|<\delta,y\in[0,1]\}} A\,\mathrm{d}\rho. \end{align} $$

2. (b) For all Borel sets $E \subseteq D_\Psi $ ,

   $$ \begin{align*} \rho(E) = \int_{-t_*}^{t_*} \rho_t(E)\,\mathrm{d} t. \end{align*} $$

Let us be as general as we can about the functions against which the conditional measures weakly converge back to the full measure. For $\alpha ,\beta \in (0,1]$ we define the following norm on continuous functions $\phi : D \to \mathcal {C}$ :

$$ \begin{align*} \| \phi \|_{\alpha;\beta} = |\phi|_{\alpha,x} + |\phi|_{\beta,y} + \|\phi\|_{L^\infty}, \end{align*} $$

where the directional Hölder semi-norms are given by

$$ \begin{align*} |\phi |_{\alpha,x} &= \sup_{x,x',y \in [0,1]} \frac{|\phi(x,y)-\phi(x',y)|}{|x-x'|^\alpha},\\ |\phi |_{\beta,y} &= \sup_{x,y,y' \in [0,1]} \frac{|\phi(x,y)-\phi(x,y')|}{|y-y'|^\beta}. \end{align*} $$

The Banach space $C^{\alpha ;\beta }$ will then consist of all continuous functions $\phi : D \to \mathcal {C}$ with $\| \phi \|_{\alpha ;\beta } < \infty $ : that is, functions that are $\alpha $ -Hölder in the x direction and $\beta $ -Hölder in the y direction. In particular, the Banach space of $C^1$ functions is continuously embedded in $C^{\alpha ;\beta }$ .

Figure 5 Picture of a baker’s map of the form (6) showing the attractor (black), an intersecting line $x = \psi (y)-t_0$ , a leaf of a transverse foliation of the form in Proposition 4.1, (purple), and the action of the map on pieces of the domain (in pale colours), and on the conditional measure of the SRB measure on the intersecting line (pink).

The following theorem, proved in Appendix B, says that for certain baker’s maps, exponential conditional mixing holds for all conditional SRB measures on a smooth foliation transversal to unstable lines:

Theorem 4.2. Suppose that $\Psi (t;y) = (\psi (y) -t,y)$ is a foliation of some subset of D for $|t| \leq t_* \in [0,1]$ and $y \in [0,1]$ , and $\psi $ is $C^2$ with $\psi ' \neq 0$ . Suppose one of the following conditions hold:

1. (I) The contractions $v_i$ are totally nonlinear^† and $C^2$ and $\bigcup _i v_i([0,1]) = [0,1]$ .

2. (II) The contractions $v_i$ are totally nonlinear and analytic, as is $\psi $ .

3. (III) The contractions are linear with $v_i(x) := \mu x + o_i$ for $o_i \in [0,1-\mu ]$ , and $\psi " \neq 0$ .

Let $\rho $ be the (unique) SRB measure of a modified baker’s map b and let $\{\rho _t\}_{t\in [-t_*,t_*]}$ be the conditional measures of $\rho $ on the foliation. Then $(b,\rho )$ has exponential conditional mixing with respect to the level set of $H(x,y) = \psi (y)-t-x$ for all $t \in (-t_*,t_*)$ .

More specifically, there exists $d^*> 0$ such that for all $\gamma \in (1-d^*,1],\beta \in (0,1], \alpha \in (\gamma -d_*,1]$ , there exist $C> 0$ , $\xi \in (0,1)$ such that for all $t \in (-t_*,t_*)$ , $A \in C^{\alpha ;\beta }$ , $B \in C^\gamma $ , $n \in \mathbb {N}$ ,

$$ \begin{align*} | \rho_t(A\circ b^n\, B) - \rho_t(B) \rho(A) | \leq C \| A \|_{C^{\alpha;\beta}} \|B\|_{C^\gamma} \xi^n. \end{align*} $$

Intuitively it seems that exponential conditional mixing holds wherever there is no obvious reason for it not to hold: that is to say, wherever the conditional measure $\mu $ is not compatible with the algebraic or symbolic structure of the unstable dynamics. Indeed, the Fourier decay results we use to prove Theorem 4.2 rely on joint non-integrability properties [22]). This incompatibility can be obtained either if $\rho _t$ is generated by nonlinear contractions (where the expanding dynamics is linear), as in I–II, or if the leaf $x = \psi (y)-t$ transforms the self-affine stable measure in a nonlinear way (as in III). We give an example where the conditional measure picks up the algebraic structure of both the stable and unstable dynamics in §4.1.

From this and Proposition 2.3, exponential convergence in Hausdorff distance of the slice sets follows.

Corollary 4.3. Under the conditions of Theorem 4.2, there exist $C_1$ and $\xi _1$ such that for all $t \in (-t_*,t_*)$ ,

$$ \begin{align*} d_{\mathrm{Haus}}(b^n(\Psi(t;\mathbb{R}) \cap \Lambda_b),\Lambda_b) \leq C_1 \xi_1^n,\end{align*} $$

where $\Lambda _b$ is the attractor of b.

Note that the classic piecewise affine baker’s maps fall under condition III of the theorem, although for conservative maps the application of Fourier dimension theory is unnecessary due to the Lebesgue-absolute continuity of $\rho $ .

It can also be seen from the proof that the constant $d_*$ is half the Fourier dimension of $\rho _t$ projected onto the x coordinate. (A choice of test functions with more specific Fourier decay properties might yield $d_*$ to be exactly the Fourier dimension.) This Fourier dimension is bounded from above by the Hausdorff dimension of $\rho _t$ , which, given the product structure of the measure, is easily seen also to be the stable dimension of the systems [23]. Thus, a larger stable dimension suggests conditional mixing holds with respect to increasingly less regular observables, with consequences for linear response theory [21, 24].

In fact, for case III, both $d^*$ and the asymptotic rate of conditional mixing $\xi $ are independent of the conditioning foliation [19, Theorem 3.1].

It is worth mentioning that it is also possible to decompose sufficiently regular (for example, real analytic) curves that are tangent to stable or unstable manifolds away from the support of the attractor into a finite set of curves that uniformly avoid tangencies. In fact, the stipulations that $\psi ' \neq 0$ or indeed that the foliation is a graph in y are mere artefacts of the proof and can almost certainly be relaxed.

To prove Theorem 4.2, we essentially use two facts. The first fact, used in Lemma B.6, is that the x component of b is a tupling map, whose action on Fourier coefficients of measures is well known. The second fact (Proposition B.3) is that the SRB measure is a product of uniform measure in the x direction and a Gibbs measure (in fact, a measure of maximal entropy) of an iterated function system in the y direction. This allows us to bring in some recent results on Fourier dimension of Gibbs measures [18, 19, 22].

The Fourier dimension of Gibbs measures is an area in progress whose results have not yet been consolidated, hence the somewhat particular set of alternatives. We remark that if $B(x,y)$ depends only on x then in case III the $\psi ' \neq 0$ restriction can be dropped: that is, quadratic tangencies with stable manifolds (lines of constant y) are allowed here.

4.1 A case where conditional mixing fails

However, it is clear that some stipulations on the contractions and foliation of a similar flavour to I–III must remain. In particular, either the contractions or the foliation should avoid preserving too good a linear structure, or conditional mixing will fail. An example where this occurs is given for the map $b_{\mathrm {bad}}$ , given by $k=3$ and

$$ \begin{align*} v_1(y) &= y/9,\\ v_2(y) &= (y+2)/9,\\ v_3(y) &= (2y+1)/3. \end{align*} $$

The attractor and dynamical picture for this map are plotted in Figure 6.

Figure 6 Picture showing the attractor $[0,1]\times C_{1/3}$ of the piecewise linear baker’s map $b_{\mathrm {bad}}$ (black) with the images of its pieces (pale colours), and the linear level set $x-y=0$ (purple). The conditional measure of the SRB measure on the intersecting line is shown in pink dots, and, in pale vertical pink lines, the $b_{\mathrm {bad}}$ -invariant subset $C_{1/3} \times [0,1]$ in which the support of the conditional measure is contained.

Here, we have the following proposition.

Proposition 4.4. $(b_{\mathrm {bad}},{\kern-1pt}\rho _{\mathrm {bad}})$ has no conditional mixing with respect to $H{\kern-1.2pt}(x{\kern-0.5pt},{\kern-1.2pt}y){\kern-1.2pt} ={\kern-1.2pt} y-x$ .

This is because the x and y directions, which are both linear, both preserve the middle-thirds Cantor set, which is copied across the directions using the linear foliation.

Proof. The contractions $v_1, v_2, v_3$ together generate the middle-thirds Cantor set $C_{1/3}$ : if $f_1(x) = x/3$ , $f_2(x) = (x+2)/3$ are the classical generators, then $v_1 = f_1^2$ , $v_2 = f_1 \circ f_2$ and $v_3 = f_2$ . The attractor $\Lambda _{b_{\mathrm {bad}}}$ of $b_{\mathrm {bad}}$ is therefore $[0,1]\times C_{1/3}$ (for example, using Proposition B.3). Our level set is $L = \{x-y=0\}$ so the support of our conditional measure is $L \cap \Lambda _{b_{\mathrm {bad}}} = \{(x,x) : x \in C_{1/3}\}$ .

The expanding dynamics of $b_{\mathrm {bad}}$ is just the tripling map $\kappa (x) = 3x\,\mod \! 1$ , which has $C_{1/3}$ as an invariant set. This means $C_{1/3} \times [0,1]$ is a proper invariant closed subset for the full baker’s map. Our conditional measure’s support $L \cap \Lambda _{b_{\mathrm {bad}}}$ is contained in this set and so for all time, $(b_{\mathrm {bad}}^n(L \cap \Lambda _{b_{\mathrm {bad}}}) \subseteq C_{1/3} \times [0,1]$ . We note that $p = (\tfrac {1}{3},\tfrac {1}{2})$ lies in $\Lambda _{b_{\mathrm {bad}}}$ , but $d(p,(b_{\mathrm {bad}}^n(L \cap \Lambda _{b_{\mathrm {bad}}}))) \geq d(p,C_{1/3}\times [0,1]) = 1/6$ , so $(b_{\mathrm {bad}}^n(L \cap \Lambda _{b_{\mathrm {bad}}})$ cannot converge in Hausdorff distance to $\Lambda _{b_{\mathrm {bad}}}$ . By the contrapositive of Proposition 2.2, no conditional mixing holds.

Nevertheless, if any nonlinear perturbation of either the level set or any of the $v_j$ is made, Theorem 4.2 will apply and conditional mixing will manifest. Conditional mixing relies on the lack of joint algebraic structure, rather than simply on regularizing properties.

5 Numerical example: Lozi map

While baker’s maps’ special structure (for example, as skew products) allows us to find rigorous results for them by borrowing existing theory, it is not immediately clear how to mathematically generalize our results. To study maps with less structure we now therefore turn to rigorously justified numerics, and consider the commonly studied and numerically amenable class of Lozi maps. These are piecewise hyperbolic affine maps $f: \mathbb {R}^2 \to \mathbb {R}^2$ with

(8) $$ \begin{align} f(x,y) = (1 + y - a|x|, bx),\quad b \neq 0. \end{align} $$

For $a \in (1,2)$ and $b \in (0,\min \{a-1,4-2a\})$ the Lozi map f has chaotic dynamics on a compact region in phase space: when additionally $b \in (0, \sqrt {2}(a - \sqrt {2}))$ this has a single mixing SRB measure [16, Theorem 5]. All unstable manifolds have positive length when $b \in (0, a - \sqrt {2})$ [25, Theorem]. A Lozi attractor is shown in Figure 1.

Lozi maps are continuous, with a jump in the Jacobian across the singularity set $\mathcal {S} = \{x = 0\}$ .

In a similar fashion to those we defined for the baker’s map (Proposition 4.1), conditional measures of the SRB measure $\mathrm {d}\rho (x,y)$ on sets $\ell _{x_0} := \{x=x_0\}$ are well defined for all $x_0 \in \mathbb {R}$ intersecting the support of the Lozi attractor [24, Theorem 2.1]. Let us denote these conditional measures by $\mathrm {d}\rho (y \mid x_0)$ . These conditional measures can be expected to have Hausdorff dimension strictly between $0$ and $1$ : in particular, they lack any manifold structure. A histogram of $\mathrm {d}\rho (y \mid 0) = \mathrm {d}\rho (y \mid (x,y) \in \mathcal {S})$ is plotted in Figure 1: the linear response for the Lozi map is determined from the mixing properties of the conditional measure on $\mathcal {S}$ [24], so we will be most interested in this particular conditional measure.

We specifically conjecture that measures $\rho (\cdot \mid x_0)$ , when pushed forward under the Lozi map f, converge back to the full SRB measure $\rho $ , and that this convergence happens at an exponential rate.

Conjecture 5.1. For generic Lozi parameters $(a,b)$ and Lebesgue-almost all $x_0 \in \mathbb {R}$ , the Lozi map has conditional mixing with respect to level curves $x = x_0$ (that is, the measures $\rho (\cdot \mid x_0)$ .

We have strong and direct numerical evidence in favour of this conjecture: Figure 7 shows exponential decay of the correlation

(9) $$ \begin{align} \int A\circ T^n\,\mathrm{d}\rho(\cdot\mid 0) - \int A\,\mathrm{d}\rho \int \mathrm{d}\rho(\cdot \mid 0) \end{align} $$

by four orders of magnitude, with reliable error quantification. In fact, it seems that A need only be piecewise $C^1$ . The consequence of Conjecture 5.1 holding on the singular line $x=0$ (up to a technical generalization to second-order mixing), as we will show in [24, Theorem 2.3], is that the Lozi map is a formal linear response to bounded dynamical perturbations [24].

Figure 7 In orange and blue, for the parameters of the Lozi map $a = 1.8$ and $b = 0.35$ , $|\rho _{\mathcal {S}}(A \circ f^n) - \rho _{\mathcal {S}}(1) \rho (A)|$ is plotted for varying n and two different observables A, where $\rho _{\mathcal {S}} = \rho (\cdot \mid 0)$ . The measure was sampled using $100$ time series of $10^7$ iterates of the segment dynamics $\vec f$ , implemented in high-precision (196-bit) validated interval arithmetic. Details of the algorithm are given in Appendix A. Two forms of error are plotted: with error bars, 99% confidence intervals for the sampling error, and with block error (not visible), the error arising from the interval arithmetic. In grey, decay of autocorrelations for a smooth observable $A(x) = 2x$ against the SRB measure, using $100$ time series of $10^8$ iterates of the full dynamics f.

It should be noted that obtaining valid samples of the quantities in (9) is tricky. To begin with, we are sampling from the conditional measure $\rho (\cdot \mid x_0)$ , which is a codimension-one object on the attractor. We then must iterate forward under the Lozi map, which is chaotic and thus unstable. To validly perform this sampling rigorously, and quantify the associated numerical error, we have developed novel algorithms, presented in Appendix A. These algorithms could, we imagine, with care be extended to general hyperbolic dynamics.

6 Conclusion

While in a numerical example, one can only consider a particular case, our results on baker’s maps suggest that conditional mixing is a very robust property. Indeed, Theorem 4.2 shows that for an open dense set of such baker’s maps (perhaps all baker’s maps), conditional mixing holds on an open dense set of analytic curves. While these skew product maps are rather special, it seems from the Fourier dimension results that conditional mixing occurs when structure is broken, and so the situation might yet be better for conditional mixing in more general maps. To this end, we make the following conjecture.

Conjecture 6.1. For all analytic Anosov diffeomorphisms on compact surfaces, conditional mixing holds on an open dense set of functions H with no critical points on their zero level set.

On the other hand, it is interesting that the exponential rates of decay for the conditional measures are substantially slower than for smooth observables against the full SRB measure: for example, in Figure 7 the rate of exponential convergence for a smooth observable $A(x,y) = x$ is much slower when the initial measure $\mu $ is a conditional measure than for the full SRB measure $\mu \sim \rho $ . Furthermore, it appears, at least for the baker’s map in some cases, that this decay rate is independent of the conditioning submanifold (see discussion in Appendix B). In a related fashion, mixing rates against the SRB measure, which depend on the essential spectrum of the transfer operator in relevant function spaces, tend to be slower for smaller Hölder regularities of observables [1]: thinking of the conditional measure $\rho (x\mid H(x) = 0)$ as equivalent to the SRB measure multiplied by a distribution $\delta (H(x))$ provides some connection between these two phenomena. Therefore, while the main connection we have seen appears to be to Fourier dimension, perhaps an appropriate functional analytic approach could yield fruit in studying this basic property of a chaotic system.

One might also ask what happens when the codimension of the conditioning submanifold is increased (in our study it has always been one). This is natural for the Bayesian filter problem since one typically makes repeated observations when observing a system [12, 20]: we expect to end up with, say, a vector of one-dimensional observations $y = (H(x), H(f(x)), H(f^2(x)),\ldots , H(f^m(x)))$ . If this dimension m is no greater than the unstable dimension (that is, the number of positive Lyapunov exponents), then the construction of the conditional measure as in Proposition 4.1 and [24, Theorem 2.1] will go through, and we might feel empowered to say that we expect conditional mixing to hold generically. On the other hand, if m is more than twice the box-counting dimension of the attractor, then the probabilistic Takens embedding theorem would tell us that y specifies x exactly $\rho $ -almost surely [5]. In the intermediate case where M lies between the unstable dimension and the attractor dimension, it could be that some kind of generic intersection property à la blenders [6] holds to give a conditional set of positive fractal dimension, which could also allow for some kind of conditional mixing. Numerical study in higher-dimensional systems may shed light on the situation.