Approximation on the limit continuum of a degenerate Kleinian group

Shirokov, N. A.

doi:10.1007/bf02367012

Cited by 1 publication

(3 citation statements)

References 4 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…If F is a quasi-Fuchsian group, then ft = ft+ U ft, ft+ M ft_ = | and for If we take into account the lemma below, the proof becomes trivial just as the proof of Theorem 1 in [2]. This lemma for X = C(A) is a classical theorem of Mergelyan azld for X = h~(A) it is contained in [2], Lemma 1. Here we give a general scheme of the proof.…”

Section: (1 )mentioning

confidence: 88%

“…The first result of this kind is obtained in [2]. It is assumed in [2] that F is degenerate, i.e., it is finitely generated with the only simply connected component ft and the limit set A which is a continuum. Let us note that the finite generation of the group F is sufficient for validity of statements on density of linear combinations of functions of the form { ~ }, a E -Z, X C ft, in some subspaees C(A) due to the following reasons.…”

Section: (1 )mentioning

confidence: 99%

“…Hence, in view of well-known problems of the approximation theory, it seems interesting for us to treat the problem of density for linear combinations of rational fractions 1 .... a E ~', in the mentioned spaces. The first result of this kind is obtained in [2]. It is assumed in [2] that F is degenerate, i.e., it is finitely generated with the only simply connected component ft and the limit set A which is a continuum.…”

mentioning

confidence: 99%

See 2 more Smart Citations

Approximation on limit compact sets of kleinian groups

Shirokov¹

1998

J Math Sci

Self Cite

View full text Add to dashboard Cite

Let F be a geometrically finite or a quasi-Fuchsian Kleiman group such that oo E fl(v). We establish the relation(1 ) X = clos x L l _ a , a E E for some countable sets E C fl(F) connected with actions of elements of F, and for the space X = C(A) or for the HSlder classes X = L~(A), 0 < a < 1, where A = A(F) = C \ fl is the limit set of F. Bibliography: 6 titles.Let F be a Kleiniangroup and let A = A(F) be its limit set. We denote ft = C\A. It is assumed o that the set A contains more than two points. We denote by ft, ~ C ft, the set of points z such that The actions of elements of the group F generate in a natural way countable sets = C ft. Hence, in view of well-known problems of the approximation theory, it seems interesting for us to treat the problem of density for linear combinations of rational fractions 1 .... a E ~', in the mentioned spaces. The first result of this kind is obtained in [2]. It is assumed in [2] that F is degenerate, i.e., it is finitely generated with the only simply connected component ft and the limit set A which is a continuum. Let us note that the finite generation of the group F is sufficient for validity of statements on density of linear combinations of functions of the form { ~ }, a E -Z, X C ft, in some subspaees C(A) due to the following reasons.Let E0 be a nowhere dense compact set such that C(Eo) # R(Eo), where R(Eo) is the closure in C(Eo) of rational functions with poles outside of E0 (for example, E0 is the famous "Swiss cheese," see [3, Chap. 2]). Then there exists an infinitely generated Kleinian group F0 such that its limit set A0 contains E0 [1, Chap. 8]. Hence, moreover, we have C(A0) r R(A0) andRecall that a Kleinian group r is called quasi-Fuchsian [1, Chap. 9] if it has exactly two invariant simply connected components separated by a Jordan curve A which is the limit set for F. A Kleinian group F is called geometrically finite [1, Chap. 6] if one can choose a fundamental polyhedron with a finite number of faces in the semispace H 3 = {(z, t) : z E C, t > 0} with a hyperbolic metric, on which action of elements of F is naturally extended. In both cases, the Kleinian group is finitely generated.Further we assume that e~ E ~t. If F is a quasi-Fuchsian group, then ft = ft+ U ft, ft+ M ft_ = | and for

show abstract

Section: (1 )mentioning

confidence: 88%

Section: (1 )mentioning

confidence: 99%

mentioning

confidence: 99%

See 1 more Smart Citation