  This paper explores the link between the ergodicity of the clus-ter
equivalence relation restricted to its infinite locus and the
indis-tinguishability of infinite clusters. It is an important element of the
dictionary connecting orbit equivalence and percolation theory. This note
starts with a short exposition of some standard material of these theories.
Then, the classic correspondence between ergodicity and in-distinguishability
is presented. Finally, we introduce a notion of strong indistinguishability
that corresponds to strong ergodicity, and obtain that this strong
indistinguishability holds in the Bernoulli case. We also define an invariant
percolation that is not insertion-tolerant, sat-isfies the Indistinguishability
Property and does not satisfy the Strong Indistinguishability Property.
