2. Definitions

A topological space Σ is called a Polish space if it is separable, and there exists a metric d on Σ which defines the topology of Σ such that $(\Sigma,d)$ is complete. Every Polish space is a Radon space. In particular, every Borel measure on Σ is regular (see [Reference Schwartz15]). From now on Σ will denote a Polish space and $\mathscr{F}$ will be the Borel σ-algebra on Σ.

A stochastic kernel on Σ is any function $K \colon \Sigma \times\mathscr{F}\to [0,1]$ such that

1. (1) the function $B\mapsto K(x,B)$, from $\mathscr{F}$ to $[0,1]$, is a probability measure for any $x\in \Sigma$;

2. (2) the function $x\mapsto K(x,B)$, from Σ to $[0,1]$, is $\mathscr{F}$-measurable for any $B\in\mathscr{F}$.

The convolution of two kernels K ₁ and K ₂ on Σ is

\begin{equation*}(K_2 \times K_1) (x,B)=\int_\Sigma K_1(x,{\rm d}y) K_2 (y,B), \qquad x \in \Sigma, B \in \mathscr{F}.\end{equation*}

The space of kernels on Σ is a convolution semigroup with identity, where the identity is the kernel $K(x,.)=\delta_x$, and δ _x stands for the Dirac measure supported on x. The iterated kernels are defined recursively, setting $K^1=K$, and for $n \geq 1$,

\begin{equation*}K^{n+1} (x,B)=\int_\Sigma K(x,{\rm d}y) K^n(y,B).\end{equation*}

A probability measure µ on $(\Sigma,\mathscr{F})$ is called K-stationary if for all $B\in\mathscr{F}$,

\begin{equation*} \mu(B)=\int K(x,B) \mu({\rm d}x). \end{equation*}

A set $B\in\mathscr{F}$ is said to be K-invariant when $K(x,B)=1$ for all $x\in B$ and $K(x,B)=0$ for all $x\in X\setminus B$. A K-stationary measure µ is called ergodic when there is no K-invariant set $B\in \mathscr{F}$ such that $0\lt\mu(B)\lt1$. As usual, ergodic measures are the extremal points in the convex set of K-stationary measures.

We say that a stochastic kernel K is strongly mixing if it admits a unique stationary probability measure µ with $\operatorname{{supp}}(\mu)=\Sigma$ and there are constants C > 0 and $0\lt\rho\lt1$ such that for every $\psi \in L^\infty(\Sigma)$, all $x\in\Sigma$ and $n\in\mathbb{N}$,

\begin{equation*} \left\vert {\int_\Sigma \psi(y)\,K^n(x,{\rm d}y) - \int_\Sigma \psi(y)\,\mu({\rm d}y) }\right\vert \leq C \rho^n \left\|{\psi}\right\|_\infty . \end{equation*}

We say that K satisfies the Doeblin condition if there is a positive finite measure ρ on $(\Sigma,\mathscr{F})$ and some ɛ > 0 such that for all $x\in\Sigma$ and $B\in\mathscr{F}$,

\begin{equation*} K(x,B)\geq 1-\varepsilon\quad \Rightarrow \quad \rho(B)\geq\varepsilon. \end{equation*}

In the context of Doeblin condition, we call the sets $\Sigma_i$, $1 \leq i \leq m$, the ergodic components of K and the integers p _i their periods. We call the sets $ \Sigma_{i,j}$, $1 \leq j \leq p_i$, the mixing subcomponents of the ergodic component $\Sigma_i$. We shall say that an ergodic component is aperiodic when its period is equal to 1.

A stochastic kernel K determines an operator $K:L^\infty(\Sigma)\to L^\infty(\Sigma)$ defined by

(1)\begin{equation} ( K \psi)(x):=\int_\Sigma \psi(y)\,K(x,{\rm d}y). \end{equation}

This operator satisfies the following.

If the kernel K on Σ is strongly mixing, we define its mixing rate to be the spectral radius of the restriction of K to the invariant subspace H ₀ in Proposition 2.2, that is,

\begin{equation*} \tau^*(K):= \rho(K\vert_{H_0}) .\end{equation*}

In general, if K satisfies the Doeblin condition, using the notation of Theorem 2.1, we define its escape rate $\beta^*(K)$ to be the spectral radius of the operator $r_\Lambda\circ K\vert_{L^\infty(\Lambda)}$, where $\Lambda=\Sigma\setminus (\Sigma_1\cup \cdots \cup \Sigma_m)$ and $r_\Lambda:L^\infty(\Sigma)\to L^\infty(\Lambda)$ is the restriction operator. Notice that the operator $K_\Lambda:L^\infty(\Lambda)\to L^\infty(\Lambda)$, $K_\Lambda= r_\Lambda\circ K\vert_{L^\infty(\Lambda)}$ is sub-stochastic and hence has spectral radius less than 1. We also define the mixing rate of the component $\Sigma_i$ as $\tau^*(K,\Sigma_i):=\sqrt[p_i]{\tau^\ast(K^{p_i},\Sigma_{i,j})} $, where $\tau^\ast(K^{p_i},\Sigma_{i,j})$ denotes the mixing rate of the strongly mixing stochastic kernel obtained by restriction of the kernel $K^{p_i}$ to any of the mixing subcomponents $\Sigma_{i,j}$, $1 \leq j \leq p_i$.

Let Σ be a discrete space, that we call a lattice. Consider a convex and compact set $X \subset \mathbb{R}^d$, a map $f \colon X \to X$, the base dynamical system and a probability kernel $K \colon \Sigma \times \Sigma \to [0,1]$. If Σ is finite with k elements, then the stochastic kernel K can be identified with a k × k stochastic matrix. We recall that a (row) stochastic matrix on a finite set Σ is any square matrix $A=[a_{ij}] \in \mathbb{R}^{k \times k}$ such that $a_{ij} \geq 0$ for all $i,j \in \Sigma$ and $\sum_{j \in \Sigma} a_{ij}=1$ for all $i \in \Sigma$. A stochastic matrix A is called primitive if there exists $n \geq 1$ for which the power matrix $A^n = [a_{ij}^{n}]$ has all entries strictly positive, that is, $a_{ij}^{n} \gt 0$ for $i, j \in \Sigma$. Notice that if A is primitive, then the kernel K on Σ determined by A is strongly mixing. In particular, Σ is an aperiodic ergodic component. We define the mixing rate of a primitive matrix A, denoted by $\tau^*(A)$, as the mixing rate of the strongly mixing kernel defined by A on Σ.

The coupled map lattice is the dynamical system $F:X^{\Sigma}\to X^\Sigma$ defined by

\begin{equation*} F (\varphi) := K( f \circ \varphi ), \end{equation*}

where $X^{\Sigma}$ denotes the space of all functions from Σ to X. Since K is a probability kernel, we get that for the constant function $\varphi = c$, the image $K (f\circ \varphi )= K f(c) = f(c)$. Hence, the set of constant functions is invariant under the dynamics defined by F.

More generally, given a Polish space Σ, consider the space $L^\infty(\Sigma,X)$ of all measurable functions $\varphi:\Sigma\to X$. Notice that since X is compact, all functions in $L^\infty(\Sigma,X)$ are bounded. We endow this space with the uniform convergence topology, which is determined by the uniform distance

\begin{equation*} d_\infty(\varphi,\psi):= \left\|{\varphi-\psi}\right\|_\infty .\end{equation*}

As before, we can define a transformation $F:L^\infty(\Sigma,X)\to L^\infty(\Sigma,X)$ by

\begin{equation*} F (\varphi) := K( f \circ \varphi ), \end{equation*}

which we will refer to as a coupled map system.

We say that the coupled map system $F:L^\infty(\Sigma,X)\to L^\infty(\Sigma,X)$ has global synchronization over a subset $\Lambda\subset \Sigma$ if for every δ > 0, there exists $n_0 \in \mathbb{N}$ such that for $n \geq n_0$ and for every $\varphi \in L^\infty(\Sigma,X)$ and all $x,y \in \Lambda$, one has

\begin{equation*}\left\|{F^n(\varphi)(x)-F^n(\varphi)(y)}\right\| \lt \delta.\end{equation*}

When $\Lambda=\Sigma$, we simply talk about global synchronization of F. Let $\Delta \subset L^\infty(\Sigma,X)$ denote the compact and convex subset of all constant functions. Notice that F has global synchronization if and only if Δ is a global attractor of F on $L^\infty(\Sigma,X)$.

We say that F has local synchronization over a subset $\Lambda\subset \Sigma$ if there exists a neighbourhood $\mathscr{U}$ of Δ in $L^\infty(\Sigma,X)$ such that for every δ > 0, there exists $n_0 \in \mathbb{N}$ such that for $n \geq n_0$ and for every $\varphi \in \mathscr{U}$ and all $x,y \in \Lambda$, one has

\begin{equation*}\left\|{F^n(\varphi)(x)-F^n(\varphi)(y)}\right\| \lt \delta.\end{equation*}

The map F has local synchronization (over Σ) if and only if Δ is a local attractor of F on $L^\infty(\Sigma,X)$.

Clearly, global synchronization implies local synchronization, but the converse is not true.

3. Main results

Throughout this section, we assume that $X\subset \mathbb{R}^d$ is a compact convex set and $f:X\to X$ is a C ¹-smooth map whose Lipschitz constant is denoted by $\operatorname{{Lip}}(f)$.

We define

\begin{equation*} \ell(f):= \lim_{n\to \infty} \frac{1}{n}\,\log \operatorname{{Lip}}(f^n). \end{equation*}

This limit exists by Fekete’s Lemma. As usual, the derivative of the map f at a point $x\in X$ is denoted by $Df_x:\mathbb{R}^d\to \mathbb{R}^d$. The top Lyapunov exponent of a periodic point $x=f^n(x)$ with period n is defined to be

\begin{equation*}\lambda(f,x):= \frac{1}{n}\,\log \left\|{D f^n_x}\right\| .\end{equation*}

By the mean value theorem, $\left\|{D f_x}\right\|\leq \operatorname{{Lip}}(f)$. Hence, for all periodic points $x\in X$ of f, one has

\begin{equation*} \lambda(f,x)\leq \ell(f).\end{equation*}

We define the oscillation semi-norm $\left\|{{.} }\right\|_{\mathrm{o}}$ of a function $\varphi \in L^\infty(\Sigma,\mathbb{R}^d)$ by

\begin{equation*}\left\|{{\varphi} }\right\|_{\mathrm{o}}:=\sup_{x,y \in \Sigma} \left\|{\varphi(x)-\varphi(y)}\right\|.\end{equation*}

Proof. Given any Lipschitz function $g \colon \mathbb{R}^d \to \mathbb{R}^d$,

\begin{align*} \left\|{{g \circ \varphi} }\right\|_{\mathrm{o}} & = \sup_{x,y \in \Sigma} \left\|{g(\varphi(x))-g(\varphi(y))}\right\| \\ & \leq \sup_{x,y \in \Sigma} \operatorname{{Lip}}(g) \left\|{\varphi(x)-\varphi(y)}\right\| \\ & \leq \operatorname{{Lip}}(g) \left\|{{\varphi} }\right\|_{\mathrm{o}}, \end{align*}

and, clearly, $\left\|{{\varphi} }\right\|_{\mathrm{o}}=0$ if and only if φ is constant.

We define the norm of a stochastic kernel K, regarded as an operator, by

\begin{equation*} \vert\!\vert\!\vert {K} \vert\!\vert\!\vert_{\mathrm{o}}:= \sup_{\left\|{{\varphi} }\right\|_{\mathrm{o}}\neq 0} \frac{\left\|{{K \varphi } }\right\|_{\mathrm{o}}}{\left\|{{\varphi} }\right\|_{\mathrm{o}}} .\end{equation*}

Proof. Given $\varphi\in L^\infty(\Sigma,X)$,

\begin{align*} \left\|{{F (\varphi) } }\right\|_{\mathrm{o}} &= \left\|{{K (f\circ \varphi) } }\right\|_{\mathrm{o}} \leq \vert\!\vert\!\vert {K} \vert\!\vert\!\vert_{\mathrm{o}} \, \left\|{{f\circ\varphi } }\right\|_{\mathrm{o}} \\ &\leq \vert\!\vert\!\vert {K} \vert\!\vert\!\vert_{\mathrm{o}}\, \operatorname{{Lip}}(f)\,\left\|{{\varphi} }\right\|_{\mathrm{o}} . \end{align*}

Hence,

\begin{equation*} \left\|{{F^n (\varphi) } }\right\|_{\mathrm{o}} \leq [ \vert\!\vert\!\vert {K} \vert\!\vert\!\vert_{\mathrm{o}} \operatorname{{Lip}}(f) ]^n\left\|{{\varphi} }\right\|_{\mathrm{o}} \end{equation*}

converges to 0 as $n\to +\infty$. This proves that F has global synchronization.

Theorem 3.3 Given a finite set Σ and a stochastic primitive matrix A on Σ, if there exists a periodic orbit $x\in X$ with Lyapunov exponent $\lambda=\lambda(f,x)$ such that

\begin{equation*}\tau^*(A)\, {\rm e}^\lambda\gt1,\end{equation*}

then F does not have local synchronization over Σ.

Theorem 3.4 Given a Polish space Σ and a stochastic kernel K on Σ satisfying the Doeblin condition with a unique aperiodic ergodic component Σ₀, if

\begin{equation*} \max\{\beta^*(K), \tau^*(K,\Sigma_0)\}\, {\rm e}^{\ell(f) }\lt1,\end{equation*}

then F has local synchronization over Σ.

The next theorem provides a sufficient condition for local synchronization over the mixing subcomponents of an ergodic component.

Theorem 3.5 Given a Polish space Σ and a stochastic kernel K on Σ satisfying the Doeblin condition with an ergodic component

\begin{equation*}\Sigma_i=\Sigma_{i,1}\cup \cdots\cup \Sigma_{i,p},\end{equation*}

where the sets $\Sigma_{i,j}$ stand for the mixing subcomponents of $\Sigma_i$, if

\begin{equation*} \tau^*(K,\Sigma_i) \, {\rm e}^{\ell(f)} \lt1,\end{equation*}

then F has local synchronization over all mixing subcomponents $\Sigma_{i,j}$ of $\Sigma_{i}$.

4. Proofs

Proof of Theorem 3.3

Let $\Delta=\{u\in X^\Sigma\colon u_i=u_j\ \forall\, i,j\in\Sigma \}$. The transformation F has local synchronization over Σ iff Δ is a local attractor whose basin of attraction is a neighbourhood of Δ. To see that F does not have local synchronization over Σ, it is enough to find the F-periodic point $u\in \Delta$ with period p such that the derivative $D F^p_u$ has an eigenvalue $\alpha\in\mathbb{C}$ with $\left\vert {\alpha}\right\vert\gt1$ associated to some (complex) eigenvector, which does not lie in the complexification of the tangent space $T_u\Delta$.

Consider the periodic point $x=f^p(x)$ whose existence is hypothesized in this theorem, and let $u=(x,x,\ldots, x)\in X^\Sigma$. Then $u\in\Delta$ is a periodic point of F with the same period p. Let $w\in\mathbb{C}^\Sigma$ be an eigenvector of A such that $\sum_{i\in\Sigma} w_i=0$ and $Aw=\beta w$, where $\left\vert {\beta}\right\vert=\tau^\ast(A)$. Writing $A=[a_{ij}]\in \mathbb{R}^{\Sigma\times \Sigma}$, let $\hat A\in (\mathbb{R}^{d\times d})^{\Sigma\times\Sigma}$ be the matrix with entries $\hat a_{ij}:= a_{ij} I_d$, where I _d is the identity matrix in $\mathbb{R}^{d\times d}$. Let $M_j= Df_{f^j(x)}$ and denote by $D_{M_j}$ the block diagonal matrix

\begin{equation*} D_{M_j}= \begin{bmatrix} M_j & 0 & \cdots & 0 \\0 & M_j & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & M_j \end{bmatrix}. \end{equation*}

Finally, let $v\in\mathbb{C}^d$ be a non-zero vector such that $D f^p_x\, v= M_{p-1}\cdots M_1\, M_0\, v= \alpha\, v$ with $\alpha\in\mathbb{C}$ and $\frac{1}{p}\,\log \left\vert {\alpha}\right\vert=\lambda(x,f)$. The Jacobian matrix of F at each point $F^j(u)$ is the matrix

\begin{equation*} D F_{F^{j}(u)} = \hat A\, D_{M_j} .\end{equation*}

Since the matrices $\hat A$ and $D_{M_j}$ commute, we have

\begin{equation*} D F^p_u = \hat{A^p} D_M, \end{equation*}

where $M= M_{p-1}\cdots M_1\, M_0=D f^p_x$. Now the complex vector $V=(w_i v)_{i\in \Sigma}\in (\mathbb{C}^d)^\Sigma$ satisfies

\begin{equation*} D F_u^p V= \hat{A^p} D_M V = \beta^p\alpha V .\end{equation*}

Since the vector V does not lie in the complexification, $\Delta_{\mathbb{C}}=\{(w,w,\ldots, w)\colon w\in\mathbb{C}\}$ of the tangent space $T_u\Delta$ and

\begin{equation*}\left\vert {\beta^p\alpha}\right\vert = {\tau^*}^{p} \, {\rm e}^{p\lambda} =( \tau^*\, {\rm e}^{\lambda} )^p \gt1,\end{equation*}

the F-invariant set Δ cannot be a local attractor.

Proof of Theorem 3.4

Assume first that the stochastic kernel K is strongly mixing, and consider the space

\begin{equation*} H_0:=\{\varphi\in L^\infty(\Sigma,\mathbb{R}^d)\colon \smallint_\Sigma \varphi \,{\rm d}\mu=0 \} \end{equation*}

introduced in Proposition 2.2. Let $\tau^*=\tau^*(K)\lt1$ be the spectral radius of $K\vert_{H_0}$.

In the end, we explain how to proceed in the general case. Taking κ > 0 with $\tau^*(K)\, {\rm e}^{\ell(f)}\lt\kappa \lt1$, there exists an integer $m\in\mathbb{N}$ such that $\vert\!\vert\!\vert {K^m} \vert\!\vert\!\vert_{\mathrm{o}}\,\operatorname{{Lip}}(f^m)\lt\kappa^m $.

Because f is $C^1-$smooth, then so is F. On the diagonal Δ, the derivatives of the linear action of K and f commute. Consequently, for all $\varphi\in\Delta$

\begin{equation*} \left\|{DF_\varphi^n}\right\| \leq {\tau^*}^n\, {\rm e}^{n\ell} \lt\kappa^n,\end{equation*}

with $0\lt\kappa\lt1$. Hence, there is a neighbourhood $\mathscr{U}$ of Δ where F ⁿ is a contraction. This implies that for all $\varphi\in \mathscr{U}$,

\begin{equation*} \lim_{k\to +\infty} \left\|{{F^{k n}\varphi} }\right\|_{\mathrm{o}} =0 . \end{equation*}

Thus, for all $0\leq j\lt n$,

\begin{equation*} \lim_{k\to +\infty} \left\|{{F^{k n+j}\varphi} }\right\|_{\mathrm{o}} =0, \end{equation*}

and therefore

\begin{equation*} \lim_{k\to +\infty} \left\|{{F^{k}\varphi} }\right\|_{\mathrm{o}}=0 . \end{equation*}

The case where K is not strongly mixing but there is only one ergodic acyclic component Σ₀ is the same as the previous one because the operator $K\vert_{H_0}$ has spectral radius equal to $\max\{\beta^*(K),\tau^*(K,\Sigma_0)\}$.

5. Coupled map classes

In this section, we introduce several classes of coupled map systems and lattices to which our results apply. In all examples below, $X\subset \mathbb{R}^d$ is any compact convex set and $f:X\to X$ is a $C^1-$smooth function.

Finite coupled map lattices

Consider a finite set $\Sigma=\{1,\ldots, k\}$ and a stochastic k × k matrix $A=[ a_{ij} ]$. The associated coupled map lattice is the transformation $F:X^k\to X^k$ with components

\begin{equation*} F_i(x) = \sum_{j=1}^k a_{ij} f(x_j), \quad \text{where } \; x=(x_1,\ldots, x_k) .\end{equation*}

In this setting, Theorem 2.1 is a simple consequence of classical Markov Chain’s Theory. Let $\Sigma_1,\ldots, \Sigma_m$ be the ergodic components of the stochastic kernel K defined by A on Σ, and let $\Lambda=\Sigma\setminus (\Sigma_1\cup \ldots \cup \Sigma_m)$. The escape rate $\beta^*(K)$ and the mixing rate $\tau^*(K,\Sigma_i)$ can be estimated as described below. See [Reference Doob5, section V-2] and also [Reference Duarte and Torres6, section 5]Footnote *. Recall that $\tau^*(K,\Sigma_i):=\sqrt[p_i]{\tau^\ast(K^{p_i},\Sigma_{i,j})} $, where $\tau^\ast(K^{p_i},\Sigma_{i,j})$ denotes the mixing rate of the strongly mixing stochastic kernel obtained by restriction of the kernel $K^{p_i}$ to any of the subcomponents $\Sigma_{i,j}$. Let $A^{p_i}=[a_{ij}^{p_i}]$. We have that

(2)\begin{equation} \beta^*(K)=\inf_{n \geq 1} [\beta(K^n)]^{1/n}, \end{equation}

where

\begin{equation*}\beta(K) =1-\min_{\ell \in \Sigma} \sum_{k \in \Sigma_1 \cup \cdots \cup \Sigma_m} a_{\ell k} = \max_{\ell \in \Sigma} \sum_{k \in \Lambda} a_{\ell k}\end{equation*}

and

(3)\begin{equation} \tau^*(K^{p_i},\Sigma_{i,j})=\inf_{n \geq 1} [\tau(K^{np_i},\Sigma_{i,j})]^{1/n}, \end{equation}

where

\begin{equation*}\tau(K^{p_i},\Sigma_{i,j}) =\frac{1}{2} \max_{\ell,\ell^\prime \in \Sigma_{i,j}} \sum_{k \in \Sigma_{i,j}} \left|a^{p_i}_{\ell k} - a^{p_i}_{\ell^\prime k} \right| = 1 - \min_{\ell,\ell^\prime \in \Sigma_{i,j}} \sum_{k \in \Sigma_{i,j}} a^{p_i}_{\ell k} \wedge a^{p_i}_{\ell^\prime k}.\end{equation*}

This class contains, for example, coupled map lattices with translation-invariant coupling and arbitrary Lipschitz continuous individual map on the periodic lattice $\mathbb{Z}_L:=\{s \in \mathbb{Z} \, \mbox{mod} \, L\}$ $(L\gt1)$.

Infinite coupled map lattices

Assume now that Σ is an infinite (countable) lattice, say $\Sigma=\mathbb{N}=\{0,1,\ldots\}$. Consider an infinite (row) stochastic matrix $A=[ a_{ij} ]$ such that for some α > 0, we have $a_{i0}\geq \alpha$ for all $i\in \mathbb{N}$. This assumptions ensure that the stochastic kernel defined by A satisfies the Doeblin condition.

The associated coupled map lattice is the transformation $F:X^{\mathbb{N}}\to X^{\mathbb{N}}$ with components

\begin{equation*} F_i(x) = \sum_{j\in\mathbb{N}} a_{ij} f(x_j) \quad \text{where } \; x=(x_j)_{j\in\mathbb{N}} .\end{equation*}

The previous finite state formulas also hold in this countable case.

Coupled map systems

Assume that Σ is a compact metric space, say $\Sigma=[0,1]$, and consider a bounded measurable function $k: [0,1]\times [0,1] \to [0,+\infty)$ such that

\begin{equation*} \int_0^1 k(x,y)\,{\rm d}y = 1 \quad \text{for all }\; x\in [0,1] . \end{equation*}

This function determines a kernel K on $[0,1]$ defined by

\begin{equation*} K(x,B)=\int_B k(x,y)\, {\rm d}y \quad \text{for all }\; B\subset [0,1] . \end{equation*}

In this setting, the stochastic kernel K satisfies the Doeblin condition.

The associated coupled map system is the transformation $F:L^\infty([0,1],X)\to L^\infty([0,1],X)$

\begin{equation*} F(\varphi)(x) = \int_0^1 k(x,y) f(\varphi(y)) \, {\rm d}y \quad \text{where } \; \varphi \in L^\infty([0,1],X).\end{equation*}

Remark 5.1. It is interesting to mention that the uncoupled map, say, the case where $F(\varphi)(x) = f(\varphi(x))$, corresponds to a dynamical system already considered in [Reference Huang and Feng8].

Remark 5.2. We note that the class of coupled map systems may contain the mean field limit of all-to-all coupled finite systems. In [Reference Sélley and Tanzi16], the authors study models of this type, with very similar dynamics to our current setup of coupled map systems.

The previous given formulas for the escape rate and the mixing rate of the ergodic components $\Sigma_i$, $1 \leq i \leq m$, extend to stochastic kernels satisfying the Doeblin condition (see [Reference Doob5, section V-5] and also [Reference Duarte and Torres6, section 5])Footnote *. We have that

\begin{equation*}\beta^*(K)=\inf_{n \geq 1} [\beta(K^n)]^{1/n},\end{equation*}

where

\begin{equation*}\beta(K) = 1-\inf_{x \in \Sigma} \int_{\Sigma_1 \cup \cdots \cup \Sigma_m} k(x,y) \,{\rm d}y = \sup_{x \in \Sigma}\int_{\Lambda} k(x,y) \,{\rm d}y, \end{equation*}

and

\begin{equation*}\tau^*(K^{p_i},\Sigma_{i,j})=\inf_{n \geq 1} [\tau(K^{np_i},\Sigma_{i,j})]^{1/n},\end{equation*}

where

\begin{equation*} \begin{array}{lcl} \tau(K^{p_i},\Sigma_{i,j}) & = & \frac{1}{2} \sup_{x,z \in \Sigma_{i,j}} \displaystyle{\int_{\Sigma_{i,j}} \left|k^{p_i}(x,y)-k^{p_i}(z,y) \right|\,{\rm d}y} \\ \\ {} & = & \displaystyle{1- \inf_{x,z \in \Sigma_{i,j}} \int_{\Sigma_{i,j}} k^{p_i}(x,y) \wedge k^{p_i}(z,y)\,{\rm d}y}. \end{array}\end{equation*}

Example 5.3. The Amari neural field equation used in biomathematics is the integro-differential equation

\begin{equation*} \partial_t u(t, x) = - u(t, x) + \int_{\mathbb{R}} {\mathcal K}(x-y) S(u(t, y)) \,{\rm d}y \qquad t \gt 0, \,x \in \mathbb{R}, \end{equation*}

where u represents the average neural activity, ${\mathcal K}$ is the connectivity kernel (modelling the interaction between neurons at distinct places) and S is the firing rate function. A first-order time discretization, considering a time step of size 1, consists in replacing the partial derivative by the difference $u(t+1, x) - u(t, x)$; we then get the map

\begin{equation*} u(t+1, x) = F( u(t, x) ) = \int_{\mathbb{R}} {\mathcal K}(x-y) S(u(t, y))\, {\rm d}y \end{equation*}

which is an example of a coupled map system where Σ is the real line (for another approximation of the Amari’s model, the reader can see [Reference Salasnich14]).

Example 5.4. Consider an integrable function $\varphi \colon \mathbb{R} \to \mathbb{R}$. The Hardy–Littlewood maximal operator maps φ to the function that is defined for any $x \in \mathbb{R}$ as

\begin{equation*} M\varphi(x) = \sup_{r \gt 0} F_r(\varphi)(x), \end{equation*}

where, for any given r > 0, F _r is given by

\begin{equation*} F_r(\varphi)(x) := \frac{1}{|B(x, r)|} \int_{B(x, r)} |\varphi(y)|\,{\rm d}y; \end{equation*}

in this expression, ${\rm d}y$ is the Lebesgue measure on $\mathbb{R}$, $B(x, r)$ is the open interval $(x-r, x+r)$ and $|B|$ denotes its Lebesgue measure. Notice that F _r corresponds to the coupled map system, where $X = \Sigma = \mathbb{R}$, $f(\cdot) = | \cdot |$ and $k(x, y) = \frac{dy}{|B(x, r)|}$ is the normalized Lebesgue measure restricted to the ball $B(x, r)$.

6. Counter-examples

In this section, we provide two counter-examples which illustrate the need for the hypothesis of the theorems as well as the limitations in their conclusions.

In the first example, the hypothesis of Theorem 3.4 is satisfied but the coupled lattice map has no global synchronization.

Example 6.1. Let $X=[0,3]$ and $f:X\to X$ be a $C^1-$smooth map such that $f(1)=2/3$, $f(8/5)=2$, $f(2)=16/9$ and

\begin{equation*}\frac{3}{2}=\left\vert {f^\prime(2)}\right\vert \lt \operatorname{{Lip}}(f)=\frac{20}{9}=\operatorname{{Lip}}\left(f\vert_{\{1,\frac{8}{5}\}}\right).\end{equation*}

An example (see Figure 1) is given by

\begin{equation*} f(t):= \begin{cases} \frac{8 \, t^3}{9}-\frac{2\, t^2}{9} & \text{for} \quad 0\leq t \lt1\\ \frac{20 (t-1)}{9}+\frac{2}{3} &\text{for} \quad 1\leq t\lt\frac{8}{5}\\ \frac{275 \, t^3}{24}-\frac{2395\, t^2}{36}+\frac{1144\, t}{9}-78 &\text{for}\quad \frac{8}{5}\leq t \lt2 \\ \frac{t^3}{18}+\frac{t^2}{3}-\frac{7\, t}{2}+7 &\text{for} \quad 2\leq t\leq 3 \end{cases}. \end{equation*}

Indeed, given $(t_0, x_0, v_0), (t_1, x_1, v_1)\in\mathbb{R}^3$ with $t_0\lt t_1$, there exists a unique cubic polynomial p(t) that interpolates these tuples in the sense that

\begin{equation*} p(t_0)=x_0,\; p^\prime(t_0)=v_0,\; p(t_1)=x_1\; \text{and } \; p^\prime(t_1)=v_1 .\end{equation*}

The above function is the unique piecewise cubic polynomial of class C ¹ that interpolates the sequence of tuples

\begin{equation*}(0,0,0),\; \left(1,\frac{2}{3}, \frac{20}{9}\right),\; \left(\frac{8}{5}, 2, \frac{20}{9}\right), \; \left(2, \frac{16}{9}, -\frac{3}{2}\right),\; (3,1, 0) .\end{equation*}

One can easily check that, in each branch, the maximum absolute value of $f^\prime(t)$ is less than or equal to $20/9$.

Figure 1.

The map f of Example 6.1.

Consider the stochastic matrix

\begin{equation*} A=\begin{bmatrix} \frac{7}{10} & \frac{3}{10} & 0 \\[4pt] 0 & \frac{7}{10} & \frac{3}{10} \\[4pt] 1 & 0 & 0 \end{bmatrix}\end{equation*}

and the coupled lattice map $F:[0,3]^3 \to [0,3]^3$ defined by

\begin{equation*}F(x,y,z)=\left(\frac{7}{10}f(x)+\frac{3}{10}f(y), \frac{7}{10}f(y)+\frac{3}{10}f(z), f(x)\right).\end{equation*}

The point $p=(8/5, 1, 2)$ is a fixed point of F. Because this fixed point is off the diagonal $\Delta=\{(x,x,x)\colon x\in [0,3] \}$, the map F cannot have global synchronization.

The matrix A has eigenvalues 1 and $\frac{1}{10}(2 \pm \sqrt{5}\,i)$ (with absolute value $3/10$). The stochastic kernel K determined by A has a unique aperiodic ergodic component $\Sigma_0=\Sigma$. Hence, $\tau^*(K,\Sigma)=3/10$ and $\beta^*(K)=0$. Consequently,

\begin{equation*}\max\{\beta^*(K), \tau^*(K,\Sigma)\} \operatorname{{Lip}}(f) \leq \frac{3}{10} \cdot\frac{20}{9} = \frac{2}{3} \lt1.\end{equation*}

The second example illustrates a transition from local synchronization over Σ to local synchronization over an ergodic component.

Example 6.2. Consider the following family of stochastic matrices

\begin{equation*}A_t:= \begin{bmatrix} \frac{1-t}{2} + \frac{4t}{9} & \frac{1-t}{2} + \frac{5t}{9}& 0 \\ \frac{1-t}{2}+\frac{t}{6} & \frac{1-t}{2}+\frac{5t}{6} & 0 \\ 0 & \frac{7(1-t)}{18} + \frac{17t}{18} & \frac{11(1-t)}{18} + \frac{t}{18} \end{bmatrix} \end{equation*}

with $0\leq t\leq 1$, whose associated family of kernels K _t has a unique aperiodic ergodic component $\Sigma_0=\{1,2\}$. Applying the formulas (2)–(3), we get $\beta^*(K_t) = \frac{11-10\,t}{18}$ and $\tau^*(K_t,\Sigma_0)=\frac{5t}{18}$.

Let $f:[0,1]\to [0,1]$ be the map $f(x):= 4x(1-x)$, which has Lipschitz constant $\operatorname{{Lip}}(f)=4$. We have ${\rm e}^{\ell(f)}= 4$ because 0 is a fixed point with derivative $f^\prime(0)=4$.

In this example, the threshold conditions of Theorems 3.4 and 3.5 are, respectively,

\begin{align*} &(\mathscr{R}_1) \quad \max\{\beta^*(K_t),\tau^*(K_t,\Sigma_0)\} \,{\rm e}^{\ell(f)} \lt1 \quad \Leftrightarrow \quad 0.65 \lt t \lt 0.9;\\ &(\mathscr{R}_2) \quad \tau^*(K_t,\Sigma_0) \,{\rm e}^{\ell(f)} \lt1 \quad \Leftrightarrow \quad t \lt 0.9. \end{align*}

An analog of the threshold condition in Theorem 3.3 with Lyapunov exponent $\lambda(f)=\log 2$ w.r.t. Lebesgue measure is

\begin{align*} (\mathscr{R}_3) \quad \max\{\beta^*(K_t),\tau^*(K_t,\Sigma_0)\}\,e^{\lambda(f)} \lt1 \quad &\Leftrightarrow \quad 0.2 \lt t . \end{align*}

See Figure 2.

Using Theorems 3.4 and 3.5,

1. (1) in region ( $\mathscr{R}_1$), (local) synchronization over Σ occurs;

2. (2) in region $(\mathscr{R}_2)$, (local) synchronization over Σ₀ occurs.

For $t\lt0.65=13/20$, an easy calculation shows that $p_t=(0,0,\frac{20 t-13}{20 t-22})$ is a fixed point of the coupled map F _t off the diagonal. This shows that there is no global synchronization for F _t on this range.

We run several numerical experiments, randomly choosing an initial condition in $[0,1]^3$, computing n = 200 iterates and then plotting the last 100 iterates. These experiments indicate that (see Figures 3 and 4):

1. (3) in region ( $\mathscr{R}_1$), global synchronization over Σ occurs;

2. (4) in region ( $\mathscr{R}_3$), local synchronization over Σ occurs;

3. (5) outside region ( $\mathscr{R}_3$), synchronization over Σ never occurs.

Figure 2.

Threshold lines.

Figure 3.

In Example 6.2, synchronization occurs, only over Σ₀, for the values $0\leq t \lt 0.2$. The right plot represents the projection of the left one onto the plane $\mathbb{R}^{\Sigma_0}$, where $\Sigma_0=\{1,2\}$.

Figure 4.

In Example 6.2, synchronization over Σ occurs for the values $0.2\lt t\leq 1$.