The dynamics of an extended, spatiotemporally chaotic system might appear extremely complex. Nevertheless, the local dynamics, observed through a finite spatiotemporal window, can often be thought of as a visitation sequence of a finite repertoire of finite patterns. To make statistical predictions about the system, one needs to know how often a given pattern occurs. Here we address this fundamental question within a spatiotemporal cat, a one-dimensional spatial lattice of coupled cat maps evolving in time. In spatiotemporal cat, any spatiotemporal state is labeled by a unique two-dimensional lattice of symbols from a finite alphabet, with the lattice states and their symbolic representation related linearly (hence ‘linear encoding’). We show that the state of the system over a finite spatiotemporal domain can be described with exponentially increasing precision by a finite pattern of symbols, and we provide a systematic, lattice Green’s function methodology to calculate the frequency (i.e., the measure) of such states.

一个扩展的、时空混乱的系统的动力学可能看起来极其复杂。尽管如此，通过有限时空窗口观察到的局部动力学通常可以被认为是有限模式的有限曲目的访问序列。为了对系统进行统计预测，需要知道给定模式出现的频率。在这里，我们解决了时空猫中的这个基本问题，时空猫是随时间演化的耦合猫图的一维空间格子。在时空猫中，任何时空状态都由来自有限字母表的唯一二维符号点阵标记，点阵状态及其符号表示线性相关（因此是“线性编码”）。