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- W2964305890 abstract "By looking at the relationship between the recurrence properties of a countable group action with a quasi-invariant measure and the structure of the space of ergodic components, we establish a simple general description of the Hopf decomposition of the action into its conservative and dissipative parts in terms of the Radon-Nikodym derivatives. As an application we describe the conservative part of the boundary action of a discrete group of isometries of a Gromov hyperbolic space with respect to an invariant quasi-conformal stream. Conservativity and dissipativity are, along with ergodicity, the most basic notions of ergodic theory which go back to its mechanical and thermodynamical origins. The famous Poincare recurrence theorem states that any measure preserving transforma- tion T of a probability space (X;m) is conservative in the sense that any positive measure subset A ‰ X is recurrent, i.e., for a.e. starting point x 2 A the trajectory fT n xg eventually returns to A. These definitions clearly make sense for any measure class preserving action G (X;m) of a general countable group G on a probability space. The opposite notions are those of dissipativity and of a wandering set, i.e., such a set A that all its translates gA; g 2 G; are pairwise disjoint. An action is called dissipative if it admits a wandering set of positive measure, and it is called completely dissipative if, moreover, there is a wandering set such that the union of its translates is (mod 0) the whole action space. Our approach to these properties is based on the observation that the notions of conservativity and dissipativity admit a very natural interpretation in terms of the ergodic decomposition of the action (under the assumption that such a decomposition exists, i.e., that the action space is a Lebesgue measure space). Let C ‰ X denote the union of all the purely non-atomic ergodic components, and let D = XnC be the union of all the purely atomic ergodic components. We call C and D the continual and discontinual parts of the action, respectively. Further, let D1 (resp., D>1) be the subset of D consisting of the points with trivial (resp., non-trivial) stabilizers, i.e., the union of free (resp., non-free) orbits in D. The restriction of the action to the set Cons = C (D>1 is conservative, whereas the restriction to the set Diss = D1 is completely dissipative, thus providing the so-called Hopf decomposition of the action space into the conservative and completely dissipative parts (Theorem 14)." @default.
- W2964305890 created "2019-07-30" @default.
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- W2964305890 date "2010-08-20" @default.
- W2964305890 modified "2023-09-27" @default.
- W2964305890 title "Hopf decomposition and horospheric limit sets" @default.
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- W2964305890 doi "https://doi.org/10.5186/aasfm.2010.3522" @default.
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