Dynamical entropy in Banach spaces

By David Kerr, Hanfeng Li. 2004

51 citations recorded by the source corpus.

We introduce a version of Voiculescu-Brown approximation entropy for isometric automorphisms of Banach spaces and develop within this framework the connection between dynamics and the local theory of Banach spaces as discovered by Glasner and Weiss. Our fundamental result concerning this contractive approximation entropy, or CA entropy, characterizes the occurrence of positive values both geometrically and topologically. This leads to various applications; for example, we obtain a geometric description of the topological Pinsker factor and show that a C*-algebra is type I if and only if every multiplier inner *-automorphism has zero CA entropy. We also examine the behaviour of CA entropy under various product constructions and determine its value in many examples, including isometric automorphisms of ℓp for 1≤p≤∞ and noncommutative tensor product shifts.

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/a32b17f9f1e48122642ed9dfd609dcfec1d5cd8a

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Applied Mathematics

1. Xiangdong Ye
2. Wen Huang
3. Eli Glasner
4. David Kerr
5. Hanfeng Li
6. Felipe Garc'ia-Ramos

In Mathematical Physics

1. Xiangdong Ye
2. Eli Glasner
3. Hanfeng Li
4. Wen Huang
5. David Kerr
6. Jian Li
7. Ke-song Yan
8. Jie-lin Li

Citations in the corpus

• Tame systems and scrambled pairs under an Abelian group action (2006, background): “Examples of tame dynamical systems include metric minimal equicontinuous systems, topologically transitive weakly almost periodic (WAP) systems (see [12]), topologically transitive hereditarily non-sensitive (HNS) systems (see [12]) and null systems under Z-action (see [20]).”
• Local entropy theory (2009, background): “(3) A null system is tame (Kerr and Li [73]).”
• Combinatorial lemmas and applications to dynamics (2009, unknown): “In the recent work by Kerr and Li [23–25] this connection was further developed.”
• Independence in topological and C*-dynamics (2006, background): “1”
• Recurrence properties and disjointness on the induced spaces (2013, result): “A remarkable result by Glasner and Weiss [8] stated that the t opological entropy of (X,T) is zero if and only if so is(M(X),TM) (similar results related to nullness and tameness can be found in [13, 14]).”
• Measure complexity and Möbius disjointness (2017, background): “Examples of tame dynamical systems include metric minimal equicontinuous systems, topologically transitive weakly almost periodic (WAP) systems (see [15]), topologically transitive hereditarily non-sensitive (HNS) systems (see [15]) and null systems (see [20, 24]).”
• Topological entropy of nonautonomous dynamical systems (2019, unknown): “Later, this connection was further developed by Kerr and Li in [12].”