Entropy range problems and actions of locally normal groups
Richard Miles Michael Björklund
Discrete & Continuous Dynamical Systems - A 2009, 25(3): 981-989 doi: 10.3934/dcds.2009.25.981
This paper deals with the problem of finding the range of entropy values resulting from actions of discrete amenable groups by automorphisms of compact abelian groups. When the acting group $G$ is locally normal, we obtain an entropy formula and show that the full range of entropy values $[0,\infty]$ occurs for actions of $G$. We consider related entropy range problems, give sufficient conditions for zero entropy and, as a consequence, verify the known relationship between completely positive entropy and mixing for these actions.
keywords: Lehmer's problem. Entropy algebraic action locally normal group entropy range
Central limit theorems in the geometry of numbers
Michael Björklund Alexander Gorodnik
Electronic Research Announcements 2017, 24(0): 110-122 doi: 10.3934/era.2017.24.012

We investigate in this paper the distribution of the discrepancy of various lattice counting functions. In particular, we prove that the number of lattice points contained in certain domains defined by products of linear forms satisfies a Central Limit Theorem. Furthermore, we show that the Central Limit Theorem holds for the number of rational approximants for weighted Diophantine approximation in $\mathbb{R}^d$. Our arguments exploit chaotic properties of the Cartan flow on the space of lattices.

keywords: Central Limit Theorems Diophantine approximation

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