Some Cohomological aspects of the Banach Fixed Point Principle

03/03/2011 - 4:00pm
03/03/2011 - 5:00pm
Speaker: 
Ludvik Janos
Abstract: 

Let $ X $ be a compact metrizable topological space and $ T : X \rightarrow X $ a continuous mapping. Let $ M(X) $ and $ C(X) $ denote the set of all compatible metrics on $ X $ and the set of all continuous real valued functions on $ X $, respectively. The mapping $ T $ is called a Banach contraction if for some $ c \in [0, 1) $ there is some metric $ d \in M (X) $ with $ d(T_x, T_y) \leq cd(x, y) $ for $ x, y \in X $. It is known that this condition is equivalent to the statement: The “core” of $ T $, i.e., $ \bigcap \{T^{n}(X): n \in N\} $, is a singleton $ \{x\} $ with $ x \in X $. We consider the set $ C(X) $ as an abelian group on which the abelian monoid $ (N,+) $ acts via $ n \cdot f = f \circ T^{n} $ for $ f \in C(X) $ and $ n \in N $. It turns out that the 1-co-cycle of this action is a mappring from N to $ C(X) $ defined by $ n \rightarrow f + 1{\cdot}f + 2{\cdot}f + \ldots + (n-1){\cdot}f $ for $ n \in N $, where $ f $ is any function from $ C(X) $. By the abuse of the language, we call $ f $ a co-cycle and, if there exists some $ g \in C(X) $ so that $ f = g - 1{\cdot}g $, we call $ f $ a coboundary. We prove that $ T $ is a Banach contraction if and only if there is a countable, point separating family $ F \subset C(X) $ of nonnegative coboundaries.

Where: 
Millikan 211, Pomona College
Misc. Information: 

Refreshments served at 3:45 p.m. at Harry Mullikin Room, Millikan 209

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