Small boundary property

In mathematics, the small boundary property is a property of certain topological dynamical systems. It is dynamical analog of the inductive definition of Lebesgue covering dimension zero.
Definition.
Consider the category of topological dynamical system ("system" in short) consisting of a compact metric space formula_1 and a homeomorphism formula_2. A set formula_3 is called small if it has vanishing orbit capacity, i.e., formula_4. This is equivalent to: formula_5 where formula_6 denotes the collection of formula_7-invariant measures on formula_1.
The system formula_9 is said to have the small boundary property (SBP) if formula_1 has a basis of open sets formula_11 whose boundaries are small, i.e., formula_12 for all formula_13.
Can one always lower topological entropy?
Small sets were introduced by Michael Shub and Benjamin Weiss while investigating the question "can one always lower topological entropy?" Quoting from their article:
"For measure theoretic entropy, it is well known and quite easy to see that a positive entropy transformation always has factors of smaller entropy. Indeed the factor generated by a two-set partition with one of the sets having very small measure will always have small entropy. It is our purpose here to treat the analogous question for topological entropy... We will exclude the trivial factor, where it reduces to one point."
Recall that a system formula_14 is called a factor of formula_9, alternatively formula_9 is called an extension of formula_14, if there exists a continuous surjective mapping formula_18 which is eqvuivariant, i.e. formula_19 for all formula_20.
Thus Shub and Weiss asked: Given a system formula_9 and formula_22, can one find a non-trivial factor formula_14 so that formula_24?
Recall that a system formula_9 is called minimal if it has no proper non-empty closed formula_7-invariant subsets. It is called infinite if formula_27.
Lindenstrauss introduced SBP and proved:
Theorem: Let formula_9 be an extension of an infinite minimal system. The following are equivalent:
Later this theorem was generalized to the context of several commuting transformations by Gutman, Lindenstrauss and Tsukamoto.
Systems with no non-trivial finite entropy factors.
Let formula_41 and formula_2 be the shift homeomorphism
This is the Baker's map, formulated as a two-sided shift. It can be shown that formula_9 has no non-trivial finite entropy factors. One can also find minimal systems with the same property.