Annales Academiæ Scientiarum Fennicæ
Mathematica
Volumen 35, 2010, 595-608

THE DIMENSION SPECTRUM OF PROJECTED MEASURES ON RIEMANN MANIFOLDS

Risto Hovila

University of Jyväskylä, Department of Mathematics and Statistics
P.O. Box 35, FI-40014 University of Jyväskylä, Finland; and
University of Helsinki, Department of Mathematics and Statistics
P.O. Box 68, FI-00014 University of Helsinki, Finland; risto.hovila 'at' helsinki.fi

Abstract. In this work we first generalize the projection results concerning the dimension spectrum of projected measures on Rⁿ to parametrized families of transversal mappings between smooth manifolds and measures on them. The projection theorems for the lower q-dimension were first considered in [FO] and [HK]. Theorems for the upper q-dimension were first considered in [FO] and [JJ]. After proving the generalized results, we compute for 1 < q \leq 2 the lower and the upper q-dimensions of the natural projection of a probability measure which is invariant under the geodesic flow on the unit tangent bundle of a two-dimensional Riemann manifold.

2000 Mathematics Subject Classification: Primary 58C35, 53D25, 28A80, 37A05, 37D40.

Key words: Projection, dimension spectrum, measure, geodesic flow.

Reference to this article: R. Hovila: The dimension spectrum of projected measures on Riemann manifolds. Ann. Acad. Sci. Fenn. Math. 35 (2010), 595-608.

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doi:10.5186/aasfm.2010.3537