Shot transition detection

2013-10-11T09:03:31Z

80.82.201.37: /* Vastness of the problem */

'''Nielsen theory''' is a branch of mathematical research with its origins in [[topological]] [[fixed point theory]]. Its central ideas were developed by Danish mathematician [[Jakob Nielsen (mathematician)|Jakob Nielsen]], and bear his name.

The theory developed in the study of the so-called ''minimal number'' of a [[map (mathematics)|map]] ''f'' from a [[compact (topology)|compact]] space to itself, denoted ''MF''[''f'']. This is defined as:
:<math>\mathit{MF}[f] = \min \{ \# \mathrm{Fix}(g) \, | \, g \sim f \},</math>
where ''~'' indicates [[homotopy]] of mappings, and #Fix(''g'') indicates the number of fixed points of ''g''. The minimal number was very difficult to compute in Nielsen's time, and remains so today. Nielsen's approach is to group the fixed point set into classes, which are judged "essential" or "nonessential" according to whether or not they can be "removed" by a homotopy.

Nielsen's original formulation is equivalent to the following:
We define an [[equivalence relation]] on the set of fixed points of a self-map ''f'' on a space ''X''. We say that ''x'' is equivalent to ''y'' if and only if there exists a [[path (topology)|path]] ''c'' from ''x'' to ''y'' with ''f''(''c'') homotopic to ''c'' as paths. The equivalence classes with respect to this relation are called the '''Nielsen classes''' of ''f'', and the '''Nielsen number''' ''N''(''f'') is defined as the number of Nielsen classes having non-zero [[fixed point index]] sum.

Nielsen proved that
:<math>N(f) \le \mathit{MF}[f],</math>
making his invariant a good tool for estimating the much more difficult ''MF''[''f'']. This leads immediately to what is now known as the '''Nielsen fixed point theorem:''' ''Any map f has at least N(f) fixed points.''

Because of its definition in terms of the [[fixed point index]], the Nielsen number is closely related to the [[Lefschetz number]]. Indeed, shortly after Nielsen's initial work, the two invariants were combined into a single "generalized Lefschetz number" (more recently called the [[Reidemeister trace]]) by [[Wecken]] and [[Reidemeister]].

==Bibliography==
*{{cite book
| last=[[Werner Fenchel|Fenchel]]
| first=[[Werner Fenchel|Werner]]
| coauthors=[[Jakob Nielsen (mathematician)|Nielsen, Jakob]]; edited by Asmus L. Schmidt
| title=Discontinuous groups of isometries in the hyperbolic plane
| series=De Gruyter Studies in mathematics
| volume=29
| publisher=Walter de Gruyter & Co.
| location=Berlin
| year=2003
}}

==External links==
*[http://at.yorku.ca/t/a/i/c/39.htm Survey article on Nielsen theory] by Robert F. Brown at [[Topology Atlas]]

[[Category:Fixed-point theorems]]
[[Category:Fixed points (mathematics)]]
[[Category:Topology]]

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