This seminar focuses on G-dynamical systems, where G stands for the groups. When G is amenable, the entropy theory and related topics (e.g., specification properties and conjugacy problems) are developed recently. However, the corresponding theory cannot be applied to the case of non-amenable groups. For example, Ornstein and Weiss exhibited an example 2-1 factor map between two non-amenable groups which increases the entropy. Such example gives a contradiction to the classical entropy theory. In this seminar we give the recent results on the entropy theory (including degree and entropy) of free (semi) groups [1-2], and extend these results to finite representation, SFT and sofic groups. Finally, we try to provide a connection to the sofic entropy theory developed by Bowen, Kerr and Li [3-5].

 

References: 

 

1. J.-C. Ban and C.-H. Chang “Tree-shifts: Irreducibility, mixing, and the chaos of tree-shifts” Trans. Amer. Math. Soc., 369.12:8389-8407, (2017).

2. J.-C. Ban and C.-H. Chang “Tree-shifts: The entropy of tree-shifts of finite type” Nonlinearity, vol. 30(7), 2785, (2017).

3. L. Bowen “A measure-conjugacy invariant for free group actions” Ann. Math. (2), 171(2):1387–1400, (2010).

4. D. Kerr and H.-F. Li “Entropy and the variational principle for actions of sofic groups” Invent. Math., 186(3):501–558, (2011).

5. D. Kerr and H.-F. Li “Soficity, amenability, and dynamical entropy” Amer. J. Math., 135(3):721–761, (2013).