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Smooth Embedding

Let \(M\) be a manifold modelled on a \(d\)-dimensional real space, \(T : M \to M\) the dynamics and \(h : M \to \mathbb{R}\) the observation. Takens' theorem [Takens1981] concerns the delay map \(\Phi_{2d+1}(x) = (h(x), h(Tx), \dots, h(T^{2d}x))\). This page follows the formal argument from the topological embedding chain to generic pairs.

The embedding chain

Proved (smoothDelayMap_isClosedEmbedding)

If \(X\) is compact, \(T\) and \(h\) are continuous and the delay map is injective, then it is a closed embedding, and a homeomorphism onto its image (smoothDelayMapRangeHomeomorph).

A continuous injection from a compact space to a Hausdorff space is a closed embedding. smoothDelayMap is kept as a name for compatibility; it is delayEmbedding (smoothDelayMap_eq_delayEmbedding).

The differential

For \(C^r\) data the delay map is \(C^r\) (contMDiff_delayEmbedding). Its differential has coordinates given by the delayed covectors

\[\ell_i(x) = Dh_{T^i x} \circ D(T^i)_x : T_x M \to \mathbb{R}, \qquad i < k\]

(delayCovector, mfderiv_delayEmbedding_apply).

Proved (injective_mfderiv_delayEmbedding_iff_span)

The delay map is an immersion at \(x\) iff the delayed covectors \(\ell_0(x), \dots, \ell_{k-1}(x)\) span the cotangent space.

Consequently an immersion needs \(k \ge d\) (finrank_le_of_injective_mfderiv_delayEmbedding), and for \(T = \mathrm{id}\) in dimension \(d \ge 2\) every covector equals \(Dh_x\), so no observation and no number of delays gives an immersion (not_injective_mfderiv_delayEmbedding_id). This is why the classical theorem is about generic pairs \((T, h)\) and not about generic \(h\) for an arbitrary fixed \(T\).

An IsContMDiffEmbedding is a \(C^r\) map with injective differentials that is a topological embedding. On a compact manifold an injective immersion is one (isContMDiffEmbedding_of_injective); the quarter turn of the circle observed by its first coordinate is a worked example, embedded by any \(k \ge 2\) delays (isContMDiffEmbedding_delayEmbedding_quarterTurn_iff).

Generic observations in a finite family

Perturb the observation inside a finite family \(\varphi_1, \dots, \varphi_N\): \(h_a = h + \sum_i a_i \varphi_i\) (perturbObservation). Its delay map is affine in \(a\).

Proved (ae_isContMDiffEmbedding_delayEmbedding_perturb)

Let \(M\) be compact and \(T\), \(h\), \(\varphi_i\) be \(C^2\), and \(k > 2d\). Suppose that along every nonzero tangent vector the differentials of the delay maps of the \(\varphi_i\) span \(\mathbb{R}^k\), and that at any two distinct points the differences of their delay vectors span \(\mathbb{R}^k\). Then for almost every \(a\) the delay map of \(h_a\) is a \(C^2\) embedding.

The proof works in extended charts, countably many by second countability. In a chart the delay map of \(h_a\) is an affine family whose derivative in \(a\) is onto by the span conditions. For immersion, the bad pairs (point, unit direction) form a set of dimension \(2d - 1 < k\); for injectivity, the bad pairs of points form a set of dimension \(2d < k\). In both cases the bad parameters are the projection of a level set of too small dimension, which is Haar-null (ae_forall_ne_of_hasStrictFDerivAt); no appeal to Sard's theorem is needed.

Where the span conditions come from

A family interpolates values at \(N\) points if any values at any \(n \le N\) distinct points are attained by a combination of it (InterpolatesValues), and interpolates derivatives similarly for directional derivatives (InterpolatesDerivatives).

  • Separation (surjective_sum_smul_sub_delayEmbedding). Let \(T\) be injective without periodic points of period at most \(2k - 2\). If the windows of \(x\) and \(y\) are disjoint, the \(2k\) points are distinct and the coordinates can be prescribed independently. Otherwise \(y = T^m x\) (or the reverse) with \(0 < m < k\), and the coordinates of the difference are \(v_j - v_{j+m}\) for the values \(v\) along one orbit segment; this triangular system has the explicit solution telescope_sub.
  • Immersion (surjective_sum_smul_mvfderiv_delayEmbedding). If \(x, \dots, T^{k-1}x\) are distinct and the differentials of \(T\) are injective, the vectors \(D(T^i)_x v\) are nonzero at distinct points, and the family prescribes the derivatives of the \(\varphi_i\) along them independently.

An explicit family on a compact smooth manifold comes from a Whitney embedding \(e : M \to \mathbb{R}^n\) (Mathlib's exists_embedding_euclidean_of_compact) and the moment functionals \(\ell_t(q) = \sum_r t^r q_r\): for \(q \ne 0\), \(t \mapsto \ell_t(q)\) is a nonzero polynomial of degree less than \(n\), so finitely many values of \(t\) suffice to avoid any given finite set of nonzero vectors (exists_forall_momentFunctional_ne_zero). The functions \(x \mapsto \ell_t(e(x))^s\) then interpolate values and derivatives by Lagrange interpolation (interpolatesValues_momentFamily, interpolatesDerivatives_momentFamily).

Proved (exists_family_forall_ae_isContMDiffEmbedding_delayEmbedding)

Takens' theorem for maps without short periodic orbits. On a compact smooth \(d\)-manifold there are finitely many smooth functions \(\varphi_q\) such that, for every injective \(C^2\) map \(T\) with injective differentials and no periodic points of period at most \(4d\), and every \(C^2\) observation \(h\), the delay map of \(h + \sum_q a_q \varphi_q\) with \(2d + 1\) coordinates is a \(C^2\) embedding for Lebesgue-almost every \(a\), in particular for some \(a\) of arbitrarily small norm (exists_family_forall_exists_isContMDiffEmbedding_delayEmbedding).

Short periodic orbits

Takens' generic maps have periodic points of small period, where every delay coordinate repeats. Two conditions on \(T\) replace the absence of such points:

  • the points of period at most \(4d\) form a countable set (for generic \(T\), a finite one);
  • at a point \(z\) of minimal period \(p \le 2d\), with \(A = D(T^p)_z\), some covector \(\omega\) detects every nonzero vector through \(\omega \circ A^q\), \(q < d\). This observability condition holds when \(A\) has \(d\) distinct eigenvalues.

Proved (exists_family_forall_ae_isContMDiffEmbedding_delayEmbedding_of_periodic)

Takens' theorem for a fixed map, in a finite family. On a compact smooth \(d\)-manifold there are finitely many smooth functions \(\varphi_q\) such that, for every injective \(C^2\) map \(T\) with injective differentials satisfying the two conditions, and every \(C^2\) observation \(h\), the delay map of \(h + \sum_q a_q \varphi_q\) with \(2d + 1\) coordinates is a \(C^2\) embedding for Lebesgue-almost every \(a\).

The proof (ae_isContMDiffEmbedding_delayEmbedding_perturb_of_periodic) splits points and pairs of points.

  • Immersion at a periodic point (exists_injective_mfderiv_delayEmbedding_perturb_of_periodic). Let \(z\) have minimal period \(p \le 2d\) and \(Q = \lfloor 2d/p \rfloor\), so that \(d \le pQ \le 2d\). The family also interpolates covectors (interpolatesCovectors_momentFamily), so the differential of the observation at \(T^r z\), \(r < p\), can be prescribed to make the delayed covector of index \(r + qp\) equal to \(\omega \circ A^{rQ + q}\). The exponents \(rQ + q\) cover \(0, \dots, d - 1\), so the differential of the delay map is injective at \(z\) for one coefficient vector. Injectivity is a polynomial condition in the coefficients, and a nonzero polynomial vanishes only on a null set (MvPolynomial.ae_eval_ne_zero); so it holds for almost every coefficient vector (ae_injective_add_sum).
  • Immersion elsewhere and separation involving an aperiodic point use the span conditions of the previous section (surjective_sum_smul_sub_delayEmbedding_of_aperiodic).
  • Pairs of periodic points are countably many, and each is separated for almost every coefficient vector by the first coordinate alone (ae_delayEmbedding_perturb_ne_of_ne).

The \(C^2\) topology and open dense observations

The space \(C^n(M, F)\) of \(C^n\) maps into a normed space carries the weak \(C^n\) topology (ContMDiffMap.instTopologicalSpace): a basic neighbourhood of \(f\) consists of the maps whose chart derivatives of order at most \(n\) are uniformly \(\varepsilon\)-close to those of \(f\) on finitely many compact sets of chart coordinates (ContMDiffMap.eventually_forall_dist_jet_lt). On a compact manifold this is the Whitney \(C^n\) topology [Hirsch1976].

Proved (exists_forall_injective_of_near)

Stability of embeddings. An injective \(C^1\) immersion of a compact manifold into a normed space has a \(C^1\) neighbourhood of injective immersions.

Near each point the chart derivative is bounded below, and a map whose chart derivative is close on a closed ball is injective there, with injective derivatives, by the mean value inequality (exists_closedBall_forall_injOn). Finitely many balls cover \(M\); the pairs of points not in a common ball form a compact set on which the map separates points by some \(\delta > 0\). Precomposition with a fixed \(C^1\) map preserves \(C^1\)-closeness (exists_forall_near_comp), so the delay map depends continuously on the observation.

Proved (isOpen_and_dense_setOf_isContMDiffEmbedding_delayEmbedding)

Takens' theorem for a fixed map, in the \(C^2\) topology. For \(T\) as above, the \(C^2\) observations whose delay map with \(2d + 1\) coordinates is a \(C^2\) embedding form an open dense subset of \(C^2(M, \mathbb{R})\).

Openness holds for every \(C^2\) map \(T\) and any number of coordinates (isOpen_setOf_isContMDiffEmbedding_delayEmbedding). Density comes from the finite family: \(h + \sum_q a_q \varphi_q \to h\) as \(a \to 0\) (ContMDiffMap.continuous_perturb), and almost every such perturbation is good (dense_setOf_isContMDiffEmbedding_delayEmbedding).

Generic pairs

The conditions on \(T\) above are those of Takens' generic diffeomorphisms. Write \(\mathrm{GoodUpTo}(T, P)\) when at every point of minimal period \(0 < p \le P\) the differential \(A = D(T^p)\) is good: \(A^m - 1\) is invertible for \(1 \le m \le 4d\) and \(A\) is observable (GoodMat). Goodness up to period \(4d\) gives both conditions (GoodUpTo.countable_periodic, GoodUpTo.observable): a nondegenerate fixed point is isolated, and on a compact manifold isolated fixed points are finitely many.

Bounded-period nondegeneracy and observability density (dense_setOf_goodUpTo), a Kupka--Smale-type lemma rather than the full Kupka--Smale theorem (no hyperbolicity or transversality of invariant manifolds), is proved by induction on the period. Let \(T\) be good up to \(P - 1\). Its points of smaller period are finitely many and isolated among the fixed points of \(T^P\); every perturbation is supported off a neighbourhood \(O\) of them, so it keeps their orbits, periods and differentials, and a \(C^0\) margin prevents new points of smaller period. The remaining fixed points of \(T^P\) have minimal period \(P\) and are covered by finitely many chart patches, each small enough that the orbit leaves it for \(P - 1\) steps. On a patch, the \(P\)-th iterate of \(S_\theta \circ T\), with \(S_\theta\) a bump perturbation \(u \mapsto u + a + L(u - c)\) near the centre, is \(S_\theta \circ T^P\). A fixed point \(u\) with linear part \(L\) determines \(a\), so the parameters with a bad fixed point are the image, under a differentiable map between spaces of equal dimension, of the pairs \((u, L)\) with \((1 + L)\, D(T^P)_u\) not good; that set is null by Fubini, since almost every \(L\) makes \((1 + L) A\) good for invertible \(A\) (ae_forall_goodMat_perturb_comp). Almost every small \(\theta\) therefore works on the patch (exists_patch), goodness on a patch persists under \(C^1\)-small perturbations (Diffeomorph.eventually_patchGood), and the patches are treated one after the other.

Proved (isOpen_and_dense_setOf_isContMDiffEmbedding_delayEmbedding_pair)

Takens' theorem for generic pairs, in the \(C^2\) topology. Let \(M\) be a compact smooth \(d\)-manifold without boundary. The pairs \((T, h)\) of a \(C^2\) diffeomorphism and a \(C^2\) observation whose delay map with \(2d + 1\) coordinates is a \(C^2\) embedding form an open dense subset of \(\mathrm{Diff}^2(M) \times C^2(M, \mathbb{R})\).

Openness is the stability of embeddings under perturbations of the pair (isOpen_setOf_isContMDiffEmbedding_delayEmbedding_pair). For density, a nonempty open set of pairs contains a product \(u \times v\) of open sets; \(u\) contains a diffeomorphism good up to period \(4d\), and for it the good observations are dense, so \(v\) contains one (dense_setOf_isContMDiffEmbedding_delayEmbedding_pair).

References

  • [Takens1981] F. Takens, Detecting strange attractors in turbulence, Lecture Notes in Mathematics 898 (1981), 366--381.
  • [SauerYorkeCasdagli1991] T. Sauer, J. A. Yorke, M. Casdagli, Embedology, J. Stat. Phys. 65 (1991), 579--616.
  • [Huke2006] J. P. Huke, Embedding nonlinear dynamical systems: a guide to Takens' theorem, MIMS EPrint 2006.26.
  • [Hirsch1976] M. W. Hirsch, Differential Topology, Graduate Texts in Mathematics 33, Springer (1976).