Coupled analytic maps.

*(English)*Zbl 0836.58027It is an important problem to determine which parts of the theory of finite-dimensional dynamical systems can be extended to infinite- dimensional ones, for example to find invariant measures for the flow. Genuinely infinite-dimensional phenomena would be expected to occur for dissipative PDE’s on unbounded domains. A class of dynamical systems is obtained by discretizing space and time and considering a recursion
\[
u(x, t+ 1)= F(x, u(\cdot, t))
\]
i.e. \(u(x, t+ 1)\), with \(x\) being a site of a lattice, is determined by the values taken by \(u\) at time \(t\). For a suitable class of \(F\)’s such dynamical systems are called coupled map lattices.

The first rigorous results on such systems are due to Bunimovich and Sinai (1988). They established the existence of an invariant measure with exponential decay of correlations in space-time in a one-dimensional lattice of weakly coupled maps. In the present paper, weakly coupled circle maps are considered and a convergent cluster expansion for the Perron-Frobenious operator is derived. Exponential mixing in space and time for an invariant SRB measure is proved. The state space of the system is \(M= (S^1)^{\mathbb{Z}^d}\). Let \(f: S\to S\) and \({\mathcal F}: M\to M\) be \[ {\mathcal F}(m)_i= f(m_i),\quad i\in \mathbb{Z}^d. \] The coupling map \(\Phi: M\to M\) is given by \[ \Phi(m)_i= m_i\exp\Biggl(2\pi i\varepsilon \sum_j g_{|i- j|}(m_i, m_j)\Biggr), \] where \(g_a:S^1\times S^1\to\mathbb{R}\). Assume the following: (A) \(f\) is expanding and real analytic. (B) \(g_a\) are exponentially decreasing and real analytic. Under these conditions, existence of the coupled map \(T\)-invariant Borel measure on \(M\) is proved, where \(T= \Phi\circ {\mathcal F}\).

The first rigorous results on such systems are due to Bunimovich and Sinai (1988). They established the existence of an invariant measure with exponential decay of correlations in space-time in a one-dimensional lattice of weakly coupled maps. In the present paper, weakly coupled circle maps are considered and a convergent cluster expansion for the Perron-Frobenious operator is derived. Exponential mixing in space and time for an invariant SRB measure is proved. The state space of the system is \(M= (S^1)^{\mathbb{Z}^d}\). Let \(f: S\to S\) and \({\mathcal F}: M\to M\) be \[ {\mathcal F}(m)_i= f(m_i),\quad i\in \mathbb{Z}^d. \] The coupling map \(\Phi: M\to M\) is given by \[ \Phi(m)_i= m_i\exp\Biggl(2\pi i\varepsilon \sum_j g_{|i- j|}(m_i, m_j)\Biggr), \] where \(g_a:S^1\times S^1\to\mathbb{R}\). Assume the following: (A) \(f\) is expanding and real analytic. (B) \(g_a\) are exponentially decreasing and real analytic. Under these conditions, existence of the coupled map \(T\)-invariant Borel measure on \(M\) is proved, where \(T= \Phi\circ {\mathcal F}\).

Reviewer: Y.Asoo (Okayama)

##### MSC:

37A99 | Ergodic theory |