450 research outputs found
The homotopy classes of continuous maps between some non-metrizable manifolds
Let R be Alexandroff's long ray. We prove that the homotopy classes of
continuous maps R^n \to R are in bijection with the antichains of P({1,...,n}).
The proof uses partition properties of continuous maps R^n \to R. We also
provide a description of for some other non-metrizable manifolds X.Comment: 17 pages, 3 figure
Kneading determinants and spectra of transfer operators in higher dimensions, the isotropic case
Transfer operators M_k acting on k-forms in R^n are associated to smooth
transversal local diffeomorphisms and compactly supported weight functions. A
formal trace is defined by summing the product of the weight and the Lefschetz
sign over all fixed points of all the diffeos. This yields a formal
Ruelle-Lefschetz determinant Det^#(1-zM). We use the Milnor-Ruelle-Kitaev
equality (recently proved by Baillif), which expressed Det^#(1-zM) as an
alternated product of determinants of kneading operators,Det(1+D_k(z)), to
relate zeroes and poles of the Ruelle-Lefschetz determinant to the spectra of
the transfer operators M_k. As an application, we get a new proof of a theorem
of Ruelle on smooth expanding dynamics.Comment: This replaces the April 2004 version: a gap was fixed in Lemma 6
(regarding order of poles) and the Axioms corrected and generalise
Long line knots
We study continuous embeddings of the long line L into L^n (n>1) up to
ambient isotopy of L^n. We define the direction of an embedding and show that
it is (almost) a complete invariant in the case n=2 for continuous embeddings,
and in the case n>3 for differentiable ones. Finally, we prove that the
classification of smooth embeddings L \to L^3 is equivalent to the
classification of classical oriented knots.Comment: 11 pages, 4 figures, to appear in Arch. Math. 82 (2004
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