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In this paper we consider families of holomorphic maps defined on subsets of the complex plane, and show that the technique developed in cite{LSvS1} to treat unfolding of critical relations can also be used to deal with cases where the critical orbit converges to a hyperbolic attracting or a parabolic periodic orbit. As before this result applies to rather general families of maps, such as polynomial-like mappings, provided some lifting property holds. Our Main Theorem states that either the multiplier of a hyperbolic attracting periodic orbit depends univalently on the parameter and bifurcations at parabolic periodic points are generic, or one has persistency of periodic orbits with a fixed multiplier.
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In this paper we will develop a very general approach which shows that critical relations of holomorphic maps on the complex plane unfold transversally in a positively oriented way. We will mainly illustrate this approach to obtain transversality for a wide class of one-parameter families of interval maps, for example maps with flat critical points, piecewise linear maps, maps with discontinuities but also for families of maps with complex analytic extensions such as certain polynomial-like maps.
We prove that any $C^{1+BV}$ degree $d geq 2$ circle covering $h$ having all periodic orbits weakly expanding, is conjugate in the same smoothness class to a metrically expanding map. We use this to connect the space of parabolic external maps (coming from the theory of parabolic-like maps) to metrically expanding circle coverings.
In this paper we will give a short and elementary proof that critical relations unfold transversally in the space of rational maps.
We show that the definition of parabolic-like map can be slightly modified, by asking $partial Delta$ to be a quasiarc out of the parabolic fixed point, instead of the dividing arcs to be $C^1$ on $[-1,0]$ and $[0,1]$.
This paper studies the linear stability problem for solitary wave solutions of Hamiltonian PDEs. The linear stability problem is formulated in terms of the Evans function, a complex analytic function denoted by $D(lambda)$, where $lambda$ is the stability exponent. The main result is the introduction of a new factor, denoted $Pi$, in the Pego-Weinstein derivative formula [ D(0) = Pi frac{dI}{dc},, ] where $I$ is the momentum of the solitary wave and $c$ is the speed. Moreover this factor turns out to be related to transversality of the solitary wave, modelled as a homoclinic orbit: the homoclinic orbit is transversely constructed if and only if $Pi eq 0$. The sign of $Pi$ is a symplectic invariant, an intrinsic property of the solitary wave, and is a key new factor affecting the linear stability. A supporting result is the introduction of a new abstract class of Hamiltonian PDEs built on a nonlinear Dirac-type equation, which model a wide range of Hamiltonian PDEs. Examples where the theory applies, other than Dirac operators, are the coupled mode equation in fluid mechanics and optics, the massive Thirring model, and coupled nonlinear wave equations. The new result is already present when the homoclinic orbit representation of the solitary wave lives in a four dimensional phase space, and so the theory is presented for this case, with the generalization to arbitrary dimension sketched.
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