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In this article we exhibit a
continuum of explicit, cubic plus flat, nonconjugate hyperbolic sectors [Theorem 3.1].
We further show that, regardless of conjugacy class, any flat vector field with a
hyperbolic sector can be locally C∞ approximated by a conjugate of a linear model;
however, there exist non-flat hyperbolic sectors which can be arbitrarily
C∞-approximated by linear conjugates, but are not conjugate to any of them
[Theorem 5.2].
In addition, we address the following:
-
Problem
- Which smooth maps can be realized as the pass-by or “sojourn time”
map of some hyperbolic sector?
Such sojourn time maps must go to ∞ at zero; the asymptotic behavior at zero
determines the sector’s conjugacy class. We prove that if τ : (0,1) → R+ is a
smooth map such that 1∕τ is smoothly extendible to zero, then there is a
smooth hyperbolic sector with τ as sojourn time map [Theorem 3.1]. In
other results, the variation in successive oscillations of τ provide sufficient
conditions for the realization of τ as a sojourn time map [Theorems 2.1 and
4.2].
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