Inducing approximations homotopic to maps between inverse limits

J. Rogers

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It is proved that there is a unique metrizable simplex $S$ whose extreme points are dense. This simplex is homogeneous in the sense that for every 2 affinely homeomorphic faces $F_{1}$ and $F_{2}$ there is an automorphism of $S$ which maps $F_{1}$ onto $F_{2}$ . Every metrizable simplex is affinely homeomorphic to a face of $S$ . The set of extreme points of $S$ is homeomorphic to the Hilbert space $ℓ_{2}$ . The matrices which represent $A (S)$ are characterized.

A special type of triangulations in numerical nonlinear analysis.

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To calculate the zeros of a map f : Rn → Rn we consider the class of triangulations of Rn so that a certain point belongs to a simplex of fixed diameter and dimension. In this paper two types of this new class of triangulations are constructed and shown to be useful to calculate zeros of piecewise linear approximations of f.

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In this paper we address the following question due to Marcy Barge: For what f:I → I is it the case that the inverse limit of I with single bonding map f can be embedded in the plane so that the shift homeomorphism $\hat{f}$ extends to a diffeomorphism ([BB, Problem 1.5], [BK, Problem 3])? This question could also be phrased as follows: Given a map f:I → I, find a diffeomorphism $F : ℝ^{2} \to ℝ^{2}$ so that F restricted to its full attracting set, $⋂_{k \geq 0} F^{k} (ℝ^{2})$ , is topologically conjugate to $\hat{f} : (I, f) \to (I, f)$ . In this situation, we say that...