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# Tuan Tran, Complexity of null dynamical systems

## Tuesday, July 4, 2023 @ 4:30 PM - 5:30 PM KST

A *theoretical dynamical system* is a pair (X,T) where X is a compact metric space and T is a self homeomorphism of X. The *topological entropy* of a theoretical dynamical system (X,T), first introduced in 1965 by Adler, Konheim and McAndrew, is a nonnegative real number that measures the complexity of the system. Systems with positive entropy are random in certain sense, and systems with zero entropy are said to be deterministic. To distinguish between deterministic systems, Huang and Ye (2009) introduced the concept of maximal pattern entropy of a theoretical dynamical system. At the heart of their argument is a Sauer-Shelah-type lemma. We will discuss this lemma and its surprising connection to a recent breakthrough in communication complexity.

Joint work with Guorong Gao, Jie Ma, and Mingyuan Rong.