CONFLUENCE OF SINGULAR POINTS OF ORDINARY DIFFERENTIAL EQUATIONS OF FUCHSIAN TYPE INDUCED BY DEFORMATION OF TWODIMENSIONAL HYPERBOLIC CONEMANIFOLD STRUCTURES CONFLUENCE OF SINGULAR POINTS OF ORDINARY DIFFERENTIAL EQUATIONS OF FUCHSIAN TYPE INDUCED BY DEFORMATION OF TWODIMENSIONAL HYPERBOLIC CONEMANIFOLD STRUCTURES
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Abstract
Let $ \{ \sigma_t \}_t \in (\infty, \infty) $ be a oneparameter family of hyperbolic Riemannian metrics on an open annulus which is continuouswith respect to the GromovHausdorff topology. We consider a system $ E_t $ of ordinary differential equations with singular points which depends on the Riemannian metric $ \sigma_t $. If $ t \neq 0 $, all of the singular points of $ E_t $ are regular. If $ t = 0 $, $ E_0 $ has an irregular singular point. In this paper, we investigate the behavior of the singular points of $ E_t $. We show that a regular singular point of $ E_t $, together with another regular singular point of $ E_t $, becomes the irregular singular point of $ E_0 $ as $ t $ $ (>0) $ tends to zero and that the irregular singular point of $ E_0 $ becomes a nonsingular point of $ E_t $ as $ t $ decreases from zero.
Let {σ<SUB><I>t</I></SUB>}<SUB><I>t</I>∈(∞,∞)</SUB> be a oneparameter family of hyperbolic Riemannian metrics on an open annulus which is continuouswith respect to the GromovHausdorff topology. We consider a system <I>E</I><SUB><I>t</I></SUB> of ordinary differential equations with singular points which depends on the Riemannian metric σ<SUB><I>t</I></SUB>. If <I>t</I> ≠ 0, all of the singular points of <I>E</I><SUB><I>t</I></SUB> are regular. If <I>t</I> = 0, <I>E</I><SUB>0</SUB> has an irregular singular point. In this paper, we investigate the behavior of the singular points of <I>E</I><SUB><I>t</I></SUB> . We show that a regular singular point of <I>E</I><SUB><I>t</I></SUB> , together with another regular singular point of <I>E</I><SUB><I>t</I></SUB> , becomes the irregular singular point of <I>E</I><SUB>0</SUB> as <I>t</I> (›0) tends to zero and that the irregular singular point of <I>E</I><SUB>0</SUB> becomes a nonsingular point of <I>E</I><SUB><I>t</I></SUB> as <I>t</I> decreases from zero.
Journal

 Kyushu Journal of Mathematics

Kyushu Journal of Mathematics 61(1), 2134, 2007
Faculty of Mathematics, Kyushu University