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 X {\displaystyle X} and a homeomorphism T : X → X {\displaystyle T:X\rightarrow X} . A set E ⊂ X {\displaystyle E\subset X} is called small if it has vanishing orbit capacity, i.e., ocap ( E ) = 0 {\displaystyle \operatorname {ocap} (E)=0} . This is equivalent to: ∀ μ ∈ M T ( X ) , μ ( E ) = 0 {\displaystyle \forall \mu \in M_{T}(X),\ \mu (E)=0} where M T ( X ) {\displaystyle M_{T}(X)} denotes the collection of T {\displaystyle T} -invariant measures on X {\displaystyle X} . The system ( X , T ) {\displaystyle (X,T)} is said to have the small boundary property (SBP) if X {\displaystyle X} has a basis of open sets { O i } i = 1 ∞ {\displaystyle \{O_{i}\}_{i=1}^{\infty }} whose boundaries are small, i.e., ocap ( ∂ O i ) = 0 {\displaystyle \operatorname {ocap} (\partial O_{i})=0} for all i {\displaystyle i} .
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 ( Y , S ) {\displaystyle (Y,S)} is called a factor of ( X , T ) {\displaystyle (X,T)} , alternatively ( X , T ) {\displaystyle (X,T)} is called an extension of ( Y , S ) {\displaystyle (Y,S)} , if there exists a continuous surjective mapping φ : X → Y {\displaystyle \varphi :X\rightarrow Y} which is eqvuivariant, i.e. φ ( T x ) = S φ ( x ) {\displaystyle \varphi (Tx)=S\varphi (x)} for all x ∈ X {\displaystyle x\in X} . Thus Shub and Weiss asked: Given a system ( X , T ) {\displaystyle (X,T)} and ε > 0 {\displaystyle \varepsilon >0} , can one find a non-trivial factor ( Y , S ) {\displaystyle (Y,S)} so that h t o p ( Y , S ) < ε {\displaystyle \operatorname {h_{top}} (Y,S)<\varepsilon } ? Recall that a system ( X , T ) {\displaystyle (X,T)} is called minimal if it has no proper non-empty closed T {\displaystyle T} -invariant subsets. It is called infinite if | X | = ∞ {\displaystyle |X|=\infty } . Lindenstrauss introduced SBP and proved: Theorem: Let ( X , T ) {\displaystyle (X,T)} be an extension of an infinite minimal system. The following are equivalent:
( X , T ) {\displaystyle (X,T)} has the small-boundary property.
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