Directional expansiveness

Expansiveness is an important property in dynamical systems related to symbolic dynamics and entropy. It was first studied in the context of continuous maps (Z-actions) and later generalized to continuous flows (R-actions). In the Z^d case, Boyle and Lind studied directional expansiveness. This followed Milnor's concept, directional properties, with the introduction of directional entropy. In the first part of the dissertation, we study expansive Z^d-actions, proving a Z^d version of the Keynes and Robertson theorem. We also find expansive directions for symbolic tensor product actions. In the second part of the dissertation, we develop a general theory of expansive directions for R^d-flows. This includes a definition of weak and strong directional expansiveness. Strong directional expansiveness implies weak directional expansiveness. The converse is false in general but holds for finite local complexity R^d-tiling dynamical systems. Our main result of the dissertation shows that the Penrose tiling dynamical system has exactly five non-strongly expansive directions. These are perpendicular to the 5th roots of unity. The proof involves using Wang tiles to show that Penrose tilings are essentially tensor products of two Sturmian dynamical systems.

Author

• Jang, Hyeeun

Language

• en

Keyword

• Penrose tilings
• Directional expansiveness
• Weak expansive
• Dynamical systems
• Strong expansive

Date created

• 2021

Type of Work

• Dissertation

License

• All rights reserved

Rights statement

• In Copyright

GW Unit

• Mathematics

Degree

• Ph.D.

Advisor

• Robinson, E.ARTHUR

Committee Member(s)

• Wu, Hao
• Sahin, Ayse
• Ullman, Daniel
• Kao, Lien-Yung (Nyima)
• Mehmetaj, Erblin

Persistent URL

•  https://scholarspace.library.gwu.edu/etd/6d56zx38q