Abstract:
This seminar focuses on Gdynamical 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 nonamenable groups. For example, Ornstein and Weiss exhibited an example 21 factor map between two nonamenable 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 [12], 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 [35].
References:
1. J.C. Ban and C.H. Chang “Treeshifts: Irreducibility, mixing, and the chaos of treeshifts” Trans. Amer. Math. Soc., 369.12:83898407, (2017).
2. J.C. Ban and C.H. Chang “Treeshifts: The entropy of treeshifts of finite type” Nonlinearity, vol. 30(7), 2785, (2017).
3. L. Bowen “A measureconjugacy 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).
