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

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Consider the category of topological dynamical system (system in short) consisting of a compact metric space   and a homeomorphism  . A set   is called small if it has vanishing orbit capacity, i.e.,  . This is equivalent to:   where   denotes the collection of  -invariant measures on  .

The system   is said to have the small boundary property (SBP) if   has a basis of open sets   whose boundaries are small, i.e.,   for all  .

Can one always lower topological entropy?

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Small sets were introduced by Michael Shub and Benjamin Weiss while investigating the question "can one always lower topological entropy?" Quoting from their article:[1]

"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   is called a factor of  , alternatively   is called an extension of  , if there exists a continuous surjective mapping   which is eqvuivariant, i.e.   for all  .

Thus Shub and Weiss asked: Given a system   and  , can one find a non-trivial factor   so that  ?

Recall that a system   is called minimal if it has no proper non-empty closed  -invariant subsets. It is called infinite if  .

Lindenstrauss introduced SBP and proved:[2]

Theorem: Let   be an extension of an infinite minimal system. The following are equivalent:

  1.   has the small-boundary property.
  2.  , where   denotes mean dimension.
  3. For every  ,  , there exists a factor   so   and  .
  4.   where   is an inverse limit of systems with finite topological entropy   for all  .

Later this theorem was generalized to the context of several commuting transformations by Gutman, Lindenstrauss and Tsukamoto.[3]

Systems with no non-trivial finite entropy factors

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Let   and   be the shift homeomorphism

 

This is the Baker's map, formulated as a two-sided shift. It can be shown that   has no non-trivial finite entropy factors.[2] One can also find minimal systems with the same property.[2]

References

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  1. ^ Shub, Michael, and B. Weiss. "Can one always lower topological entropy?." Ergodic Theory and Dynamical Systems 11.3 (1991): 535–546.
  2. ^ a b c Lindenstrauss, Elon (1999-12-01). "Mean dimension, small entropy factors and an embedding theorem". Publications Mathématiques de l'Institut des Hautes Études Scientifiques. 89 (1): 227–262. doi:10.1007/BF02698858. ISSN 0073-8301.
  3. ^ Gutman, Yonatan, Elon Lindenstrauss, and Masaki Tsukamoto. "Mean dimension of  -actions." Geometric and Functional Analysis 26.3 (2016): 778–817.