000079 Dani S G (School of Mahematics, Tata Inst of Fundamental Res, Homi Bhabha Rd, Colaba, Mumbai-400 005, Email: dani@math.tifr.res.in) : Dynamical properties of linear and projective transformations and their applications. Indian J pure appl Math 2004, 35(12), 1365-94.
Let be a finite-dimensional vector space over the field IR of real numbers, namely IRd for some d≥O. We denote by P(V) the corresponding projective space; thus P(V)= (V{0})<126>, where <126> is the equivalence relation identifying every υ<198>V{0} with tυ for all t <198> IR* (nonzero real numbers). We consider V equipped with the usual Euclidean topology and the projective space equipped with the quotient topology as the space of equivalence classes. Let T : V→ V be a nonsingular linear transformation of V. Then T carries every equivalence class in V{0}, with respect to <126> as above, to an equivalence class and hence induces a homeomorphism of P(V); we call this the projective transformation associated with T. The nonsingular linear transformations and projective transformations have been considered as dynamical systems. Their dynamical properties, and their applications to certain topics in Lie groups and ergodic theory have been designed. Borel's density theorem, Halmos's question on existence of ergodic automorphisms, invariant measures of automorphisms of locally compact groups etc. This is primarily an expository article, but some new points have been brought out, corrections to some arguments in Dani are noted, and some questions are raised. While for the most part only real vector spaces, and their analogues over complex numbers have been considered, analogues of the results in the case of vector spaces over p-adic fields have been briefly described.
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