scholarly journals DISCRETENESS CRITERIA FOR MÖBIUS GROUPS ACTING ON II

2009 ◽  
Vol 80 (2) ◽  
pp. 275-290 ◽  
Author(s):  
LIU-LAN LI ◽  
XIAN-TAO WANG
Keyword(s):  
Fixed Point ◽  
Necessary Conditions ◽  
Careful Analysis ◽  
Necessary Condition ◽  
Sufficient Condition ◽  
Point Sets ◽  
Discreteness Criteria ◽  
Möbius Groups ◽  
Jørgensen’S Inequality ◽  
Fixed Point Sets

AbstractJørgensen’s famous inequality gives a necessary condition for a subgroup of PSL(2,ℂ) to be discrete. It is also true that if Jørgensen’s inequality holds for every nonelementary two-generator subgroup, the group is discrete. The sufficient condition has been generalized to many settings. In this paper, we continue the work of Wang, Li and Cao (‘Discreteness criteria for Möbius groups acting on $\overline {\mathbb {R}}^n$’, Israel J. Math.150 (2005), 357–368) and find three more (infinite) discreteness criteria for groups acting on $\overline {\mathbb {R}}^n$; we also correct a linguistic ambiguity of their Theorem 3.3 where one of the necessary conditions might be vacuously fulfilled. The results of this paper are obtained by using known results regarding two-generator subgroups and a careful analysis of the relation among the fixed point sets of various elements of the group.

scholarly journals Product involutions with0or1-dimensional fixed point sets on orientable handlebodies

1994 ◽  
Vol 17 (3) ◽  
pp. 457-462
Author(s):  
Roger B. Nelson
Keyword(s):  
Fixed Point ◽  
Necessary Conditions ◽  
Point Sets ◽  
Fixed Point Set ◽  
Point Set ◽  
Fixed Point Sets

Lethbe an involution with a0or1-dimensional fixed point set on an orientable handlebodyM. We show that obvious necessary conditions for fiberingMasA×Iso thath=τ×rwithτan involution ofAandrreflection about the midpoint ofIalso turn out to be sufficient. We also show that such a “product” involution is determined by its fixed point set.

Fixed point sets for abstract volterra operators on fréchet spaces

2000 ◽  
Vol 76 (1-2) ◽  
pp. 131-152 ◽  
Author(s):  
Dana M. Bedivan ◽  
Donal O′Regan
Keyword(s):  
Fixed Point ◽  
Fréchet Spaces ◽  
Point Sets ◽  
Volterra Operators ◽  
Fixed Point Sets

scholarly journals Syarat Perlu dan Cukup untuk Keterbatasan Potensial Riesz di Ruang Morrey Klasik

2013 ◽  
Vol 14 (3) ◽  
pp. 227
Author(s):  
Mohammad Imam Utoyo ◽  
Basuki Widodo ◽  
Toto Nusantara ◽  
Suhariningsih Suhariningsih
Keyword(s):  
Characteristic Function ◽  
Morrey Space ◽  
Necessary Conditions ◽  
Riesz Potential ◽  
Necessary Condition ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Research Results ◽  
Necessary And Sufficient

This script was aimed to determine the necessary conditions for boundedness of Riesz potential in the classical Morrey space. If these results are combined with previous research results will be obtained the necessary and sufficient condition for boundedness of Riesz potential. This necessary condition is obtained through the use of characteristic function as one member of the classical Morrey space.

scholarly journals Fixed poin sets in digital topology, 1

2020 ◽  
Vol 21 (1) ◽  
pp. 87 ◽  
Author(s):  
Laurence Boxer ◽  
P. Christopher Staecker
Keyword(s):  
Fixed Point ◽  
Fixed Points ◽  
Digital Images ◽  
Fixed Point Theory ◽  
Point Theory ◽  
Point Sets ◽  
Fixed Point Set ◽  
Complete Computation ◽  
Point Set ◽  
Fixed Point Sets

<p>In this paper, we examine some properties of the fixed point set of a digitally continuous function. The digital setting requires new methods that are not analogous to those of classical topological fixed point theory, and we obtain results that often differ greatly from standard results in classical topology.</p><p>We introduce several measures related to fixed points for continuous self-maps on digital images, and study their properties. Perhaps the most important of these is the fixed point spectrum F(X) of a digital image: that is, the set of all numbers that can appear as the number of fixed points for some continuous self-map. We give a complete computation of F(C<sub>n</sub>) where C<sub>n</sub> is the digital cycle of n points. For other digital images, we show that, if X has at least 4 points, then F(X) always contains the numbers 0, 1, 2, 3, and the cardinality of X. We give several examples, including C<sub>n</sub>, in which F(X) does not equal {0, 1, . . . , #X}.</p><p>We examine how fixed point sets are affected by rigidity, retraction, deformation retraction, and the formation of wedges and Cartesian products. We also study how fixed point sets in digital images can be arranged; e.g., for some digital images the fixed point set is always connected.</p>

Biconnected Multifunctions of Trees which have an end Point as Fixed Point or Coincidence

1971 ◽  
Vol 23 (3) ◽  
pp. 461-467 ◽  
Author(s):  
Helga Schirmer
Keyword(s):  
Fixed Point ◽  
Monotone Mapping ◽  
Monotone Mappings ◽  
Point Sets ◽  
End Point ◽  
Continuous Use ◽  
Fixed Point Sets

It was proved almost forty years ago that every mapping of a tree into itself has at least one fixed point, but not much is known so far about the structure of the possible fixed point sets. One topic related to this question, the study of homeomorphisms and monotone mappings of trees which leave an end point fixed, was first considered by G. E. Schweigert [6] and continued by L. E. Ward, Jr. [8] and others. One result by Schweigert and Ward is the following: any monotone mapping of a tree onto itself which leaves an end point fixed, also leaves at least one other point fixed.It is further known that not only single-valued mappings, but also upper semi-continuous (use) and connected-valued multifunctions of trees have a fixed point [7], and that two use and biconnected multifunctions from one tree onto another have a coincidence [5].

scholarly journals Fixed point sets of isotopies on surfaces

2020 ◽  
Vol 22 (6) ◽  
pp. 1971-2046
Author(s):  
François Béguin ◽  
Sylvain Crovisier ◽  
Frédéric Le Roux
Keyword(s):  
Fixed Point ◽  
Point Sets ◽  
Fixed Point Sets
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