 Bernoulli systems mark the highest level in EH. In order to define Bernoulli systems we first have to introduce the notion of a partition of \(X\) (sometimes also called the ‘coarse graining of \(X\)’). A partition of X is a division of \(X\) into different parts (the so-called ‘atoms of the partition’) so that these parts don’t overlap and jointly cover \(X\) (i.e., they are mutually exclusive and jointly exhaustive). For instance, in Figure 1 there is a partition of the phase space that has two atoms (the left and the right part). More formally, \(\alpha = \{\alpha_1,\ldots , \alpha_n\}\) is a partition of \(X\) (and the \(\alpha_i\) its atoms) iff (i) the intersection of any two atoms of the partition is the empty set, and (ii) the union of all atoms is \(X\) (up to measure zero). Furthermore it is important to notice that a partition remains a partition under the dynamics of the system. That is, if \(\alpha\) is a partition, then \(T_n\alpha = \{T_n\alpha_1 ,\ldots ,T_n\alpha_n\}\) is also a partition for all \(n\).