Abstract
In this chapter we gently introduce the reader to hyperbolicity theory in dynamical systems, and relate normal hyperbolicity to hyperbolic fixed points and center manifolds. Then we give some basic examples and motivation for studying the noncompact case. We give a brief overview of the history and literature and compare the two methods of proof in the basic setting of a hyperbolic fixed point. Then we continue to introduce the concept of bounded geometry and a precise statement of the main result of this work and discuss its relation to the literature. We describe a few extensions and details of the results and conclude the chapter with notation used throughout this book.
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Notes
- 1.
For simplicity of presentation we ignore the facts that \(\Phi \) may have a smaller domain of definition, or that it is a semi-flow or semi-cascade, only defined on \(T \ge 0\).
- 2.
I do not know whether loss of smoothness is generic for NHIMs. See [Has94, HW99] for the case of Anosov systems.
- 3.
We remove the eigenvalues associated to \(E_0\) from \(E_\pm \) so that \(E_-,\,E_0,\,E_+\) together disjointly span the total tangent space at \(x\).
- 4.
The definition of normal hyperbolicity in [Ma\(\tilde{\mathrm{n}}\)78] is a bit more general than the definition in this paper. That definition only requires a growth ratio \(r\ge 1\) along solution curves in the invariant manifold, and not as a ratio of global growth rates \(\rho _{\scriptscriptstyle X}, \rho _{\scriptscriptstyle Y}\), see also Remark 1.10.
- 5.
The case \(r= \infty \) would require \(\rho _{\scriptscriptstyle X}> 0\); when \(\rho _{\scriptscriptstyle X}= 0\), any finite order \(r\) can be obtained, but only for perturbations sufficiently small depending on \(r\).
- 6.
The ratio \(r\) in the spectral gap is defined by a strict inequality, which we ignore here for simplicity of presentation.
- 7.
Here, a distribution is meant in the sense of differential geometry as a subbundle of the tangent bundle, not a generalized function (nor a probability distribution).
- 8.
These facts were pointed out to me by Duistermaat.
- 9.
Henry actually has reversed notation where the ‘vertical’ manifold \(Y \times \{0\}\) is the NHIM.
- 10.
Their Hypothesis (H2) that a certain approximate splitting like (1.9) “does not twist too much”, can be obtained from uniform Lipschitz continuity of the tangent spaces of the invariant manifold. I am not sure if this is a significantly weaker hypothesis. See also the discussion in Remark 3.13.
- 11.
Contrary to the graph transform (which is only intrinsically defined for mappings), the Perron method can be formulated both for flows and discrete mappings. For the discrete case, the integral must be replaced by a sum, the mapping \(\Phi \) must be split into a linear and nonlinear part, and the linearized flow must be replaced by iterates of the linearized mapping. See for example [APS02; PS04].
- 12.
This is for the continuous case. The imaginary axis of the spectrum of a vector field corresponds (via the exponential map) to the unit circle for the spectrum of a diffeomorphism in the discrete case.
- 13.
The term ‘fixed point’ in the context of a non-autonomous system is not definable in a coordinate-free way: any orbit of the system can be made into a fixed point under a suitable time-dependent coordinate transformation. However, there may be a preferred “time-independent” coordinate system. Moreover, the hyperbolicity of an orbit with respect an intrinsic metric is independent of a choice of coordinates.
- 14.
We do not claim that bounded geometry is a necessary condition to generalize the theory of normal hyperbolicity to noncompact ambient spaces, only that it is sufficient. Section 3.3 does contain some examples, though, that indicate that some form of bounded geometry is necessary.
- 15.
We actually only obtain \(C^k\) closeness for integer \(k \le r- 1\) where \(r\) is the ratio in the spectral gap condition (1.11). This is probably an artifact of the techniques we used, while \(C^{k,\alpha }\) closeness with \(k + \alpha = r\) should be obtainable.
- 16.
Note that expansion along \(E^+\) could also be formulated as \({\left||\mathrm{{D}}\Phi ^t(m)\,x\right||} \ge C_+\,e^{\rho _+\,t}\,{\left||x\right||}\) for \(t \ge 0\) and \((m,x) \in E^+\). This is equivalent to the condition as stated, which says that there is contraction for \(t \le 0\), that is, in backward time. This latter formulation is preferable because it is the form required in estimates.
- 17.
If the perturbation is time-dependent, but in an (almost) periodic way, then this can still be treated in the compact setting. One can extend the configuration space with the circle \(S^1\) (or an \(n\)-torus in the almost periodic case).
- 18.
The order of contact is defined as the degree up to and including which the Taylor expansions of the objects agree.
- 19.
The number of preimages must be countable if \(M\) is assumed to be second-countable.
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Eldering, J. (2013). Introduction. In: Normally Hyperbolic Invariant Manifolds. Atlantis Series in Dynamical Systems, vol 2. Atlantis Press, Paris. https://doi.org/10.2991/978-94-6239-003-4_1
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DOI: https://doi.org/10.2991/978-94-6239-003-4_1
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