1. Let be the doubling map on the unit torus. We also consider the uneven doubling for and for . It is easy to see that the Lebesgue measure is -invariant, ergodic and the metric entropy . In particular, and when .

2. Following is a theorem of Einsiedler–Fish here.

**Proposition.** Let be the doubling map on the unit torus, be an -invariant measure with zero entropy. Then for any , , there exist and a subset with , such that for all , and all : .

A trivial observation is , which also follows from general entropy-dimension formula.

**Proof.** Let and be fixed. Consider the generating partition , and its refinements (separated by )….

Furstenberg introduced the following notation in 1967

**Definition.** A multiplicative semigroup is *lacunary*, if for some integer . Otherwise, is *non-lacunary*.

**Example.** Both and are lacunary semigroups. is a non-lacunary semigroup.

**Theorem.** Let be a non-lacunary semigroup, and enumerated increasingly by . Then .

**Example.** . It is equivalent to show has smaller and smaller steps.

**Theorem.** Let be a non-lacunary semigroup, and be -invariant. If is not isolated in , then .

**Furstenberg Theorem.** Let be a non-lacunary semigroup, and . Then is dense in .

In the same paper, Furstenberg also made the following conjecture: a -invariant ergodic measure is either supported on a finite orbit, or is the Lebesgue measure.

A countable group is said to be amenable, if it contains at least one Følner sequence. For example, any abelian countable group is amenable. Note that for amenable group action , there always exists invariant measures and the decomposition into ergodic measures. More importantly, the generic point can be defined by averaging along the Følner sequences, and almost every point is a generic point for an ergodic measure. In a preprint, the author had an interesting idea: to prove Furstenberg conjecture, it suffices to show that every irrational number is a generic point of the Lebesgue measure. Then any other non-atomic ergodic measures, if exist, will be starving to death since there is no generic point for them 🙂