Examples The simplest example is when is a finite set and the counting measure. Then a self-map of preserves if and only if it is a bijection, and it is ergodic if and only if has only one orbit (that is, for every there exists such that ). For example, if then the cycle is ergodic, but the permutation is not (it has the two invariant subsets and ). Equivalent formulations The definition given above admits the following immediate reformulations: for every with we have or (where denotes the symmetric difference); for every with positive measure we have ; for every two sets of positive measure, there exists such that ; Every measurable function with is constant on a subset of full measure. Importantly for applications, the condition in the last characterisation can be restricted to square-integrable functions only: If and then is constant almost everywhere. Further examples Bernoulli shifts and subshifts Let be a finite set and with the product measure (each factor being endowed with its counting measure). Then the shift operator defined by is . There are many more ergodic measures for the shift map on . Periodic sequences give finitely supported measures. More interestingly, there are infinitely-supported ones which are subshifts of finite type. Irrational rotations Let be the unit circle , with its Lebesgue measure . For any the rotation of of angle is given by . If then is not ergodic for the Lebesgue measure as it has infinitely many finite orbits. On the other hand, if is irrational then is ergodic. Arnold's cat map Let be the 2-torus. Then any element defines a self-map of since . When one obtains the so-called Arnold's cat map, which is ergodic for the Lebesgue measure on the torus. Ergodic theorems If is a probability measure on a space which is ergodic for a transformation the pointwise ergodic theorem of G. Birkhoff states that for every measurable functions and for -almost every point the time average on the orbit of converges to the space average of . Formally this means that The mean ergodic theorem of J. von Neumann is a similar, weaker statement about averaged translates of square-integrable functions. Related properties Dense orbits An immediate consequence of the definition of ergodicity is that on a topological space , and if is the σ-algebra of Borel sets, if is -ergodic then -almost every orbit of is dense