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In the case [MATH] , the rank of [MATH] is in fact [MATH] or greater. To verify, suppose [MATH] Assume [MATH] , where [MATH] are eigenvectors at the periodic point [MATH] . Assume [MATH] where [MATH] are eigenvectors at point [MATH] obtained by pushing [MATH] along with the map [MATH] . Likewise, if [MATH] is of period... |
Then the compressed form of [MATH] is [MATH] with [EQUATION] where [MATH] are characteristic multipliers and the pattern is repeated until [MATH] rows are obtained, and |
[EQUATION] The rank of [MATH] is equal to the number of its rows by Lemma . The rank of [MATH] is [MATH] because the Vandermonde matrix |
[EQUATION] has full rank, the [MATH] being distinct by assumption. Therefore, the rank of [MATH] is [MATH] or greater for each [MATH] |
To complete the proof, we note that [MATH] is of dimension [MATH] for [MATH] and that [MATH] is of dimension [MATH] A new assumption about [MATH] is useful. |
Assumption about [MATH] (3): It is assumed that [MATH] can be covered with [MATH] [MATH] -balls for [MATH] . It is assumed that [MATH] and therefore [MATH] |
can be covered with [MATH] [MATH] -balls. We also extend the assumption about the Lipshitz bound [MATH] Assumption about [MATH] (2): It is assumed that the Lipshitz constant of [MATH] with respect to [MATH] for [MATH] |
is upper bounded by [MATH] . This assumption too may be verified using compactness like the first assumption about [MATH] The proof may now be completed easily. Suppose [MATH] |
for some [MATH] . Then [MATH] at the center of one of the [MATH] balls covering [MATH] . By the transfer of volume Lemma the probability of such an event is upper bounded by |
[EQUATION] where [MATH] over [MATH] The probability evidently goes to [MATH] as [MATH] leaving us with a measure zero set of [MATH] where [MATH] |
is not immersive at some point in [MATH] for [MATH] The sets [MATH] and [MATH] are handled similarly. Theorem assumes [MATH] to be a closed and compact submanifold. That assumption implies [MATH] |
to be compact. If [MATH] is compact, we may conclude that [MATH] over [MATH] exists and is positive. The assumptions on [MATH] can be reduced. However, the technicalities that arise (see |
) are extraneous to the main ideas in this paper. Perturbing the dynamical system Let [MATH] be a diffeomorphism, which is as before but with [MATH] . Let [MATH] denote [MATH] The vector in [MATH] with first component [MATH] and the others zero is denoted by [MATH] . The perturbed dynamical system is |
[EQUATION] with [MATH] , where [MATH] is the embedding dimension. It may be noted we are only perturbing the first coordinate of [MATH] Because the observation function will be assumed to be [MATH] it is enough to perturb only the first coordinate. |
The delay vector under [MATH] is [EQUATION] Convention about [MATH] : It is assumed that [MATH] . Thereafter, it is assumed that [MATH] |
[MATH] and so on. The delay vector under [MATH] is therefore [EQUATION] It is worthy of notice that [MATH] perturbs only the first component of [MATH] . Because the delay vector is built up using [MATH] |
[MATH] must perturb the first component. If not, the perturbation may not propagate to the delay vector at all. It turns out that perturbing only the first component is also sufficient to obtain a prevalence theorem. |
Our first task is to express [MATH] as a perturbation of [MATH] That can be done by simply iterating the definition of [MATH] [EQUATION] |
Above and later, [MATH] is the same as [MATH] By following the pattern, we obtain [EQUATION] for [MATH] . Here it is important to note that [MATH] |
is linear in [MATH] . For brevity, we will rewrite ( 5.1 ) as [EQUATION] We then get [EQUATION] with the matrix [MATH] defined by |
[EQUATION] The next lemma is about the rank of [MATH] Lemma 10 If [MATH] are distinct, the rank of [MATH] is equal to the number of its rows. |
Proof. Suppose we consider [EQUATION] The rank lemma tells us that the rank of [MATH] is equal to the number of its rows. Now to produce a vector [MATH] |
in the range of [MATH] , we proceed as follows. Define [EQUATION] and so on. Because of the linearity of [MATH] in [MATH] the vector [MATH] that satisfies [MATH] |
also satisfies [MATH] The next lemma is similar. Part (c) of the following lemma is more general than Lemma 10 because we allow [MATH] |
Lemma 11 The following matrices have rank equal to the number of rows: [MATH] assuming [MATH] to be distinct. [MATH] where [MATH] is the first [MATH] rows of [MATH] assuming [MATH] to be distinct. |
[MATH] assuming [MATH] are distinct and [MATH] Proof. Similar to the previous proof. Our second task in this section is to obtain [MATH] |
as a perturbation of [MATH] . It is helpful to introduce another convention: Convention about [MATH] [MATH] [MATH] is obtained as |
[MATH] [MATH] is obtained as [MATH] and so on. Thus, in effect we need to obtain perturbative expansions of [MATH] To do so, let us first note that |
[EQUATION] We substitute the above equation into the iteration that defines [MATH] and obtain [EQUATION] and so on. If we now use ( 5.1 ) to substitute for [MATH] , we obtain |
[EQUATION] where [MATH] is linear in [EQUATION] We may then write [EQUATION] where [EQUATION] The second task for this section concludes with a lemma about the rank of [MATH] |
Lemma 12 If [MATH] are distinct, the rank of [MATH] is equal to the number of its rows. Proof. The proof is similar to that of Lemma 10 . First consider |
[EQUATION] By Lemma , the rank of this matrix is equal to the number of its rows. Suppose we want to find [MATH] such that [MATH] equals a specified vector |
[MATH] . To do so, we find a vector [MATH] such that the matrix displayed above applied to [MATH] is equal to [EQUATION] where [MATH] [MATH] where [MATH] is [MATH] evaluated by replacing |
[MATH] by [MATH] and [MATH] by [MATH] and so on. The third and final task of this section is to track the perturbation of fixed points when the map [MATH] is perturbed to [MATH] |
Lemma 13 Suppose [MATH] and [MATH] has no eigenvalue equal to [MATH] Under [MATH] , the fixed point [MATH] perturbs to [EQUATION] |
Proof. The function [MATH] exists by the implicit function theorem. To obtain the expansion given in the lemma, start with [EQUATION] |
differentiate with respect to [MATH] and obtain [MATH] at [MATH] using implicit differentiation. The setting for injectivity and immersivity theorems |
In the case where [MATH] is fixed and only the observation function [MATH] is perturbed, injectivity and immersivity are proved with respect to the ball [MATH] , where [MATH] can be anything. Such a thing is plainly impossibly when [MATH] is perturbed to [MATH] . Under a perturbation, the map may even fail to be well d... |
We will assume that [MATH] is a compact sphere in [MATH] centered at the origin. The map [MATH] will be proved to be injective and immersive over [MATH] . It is assumed that [MATH] |
is a compact sphere bigger than [MATH] and containing [MATH] . If [MATH] it is assumed that [MATH] all remain in [MATH] . In addition, [MATH] is assumed to be so small that [MATH] |
all remain in [MATH] for all [MATH] . Further assumptions are enumerated below: 1. [MATH] is assumed to be a diffeomorphism (for [MATH] ), that is [MATH] or better. |
2. The map [MATH] has exactly [MATH] fixed points and those will be denoted by [MATH] 3. The map [MATH] has no other periodic points of period less than [MATH] |
4. All the fixed points are hyperbolic and [MATH] if [MATH] . This assumption is made with the intention of simplifying the proof so as to bring out the main techniques with greater clarity. Here we are essentially assuming injectivity between fixed points. |
5. We will also assume that [MATH] is immersive at each fixed point for the same reason. Now we will recall a few basic facts about Lebesgue points. A point |
[MATH] is a Lebesgue point of a measurable set [MATH] if [EQUATION] We will need the following basic lemma. Lemma 14 If every point of the measurable set [MATH] is a Lebesgue point of the measurable set [MATH] , then [MATH] |
Proof. Almost every point of [MATH] is a Lebesgue point of [MATH] Similarly, almost every point of [MATH] , the complement of [MATH] is a Lebesgue point of [MATH] . If [MATH] is a Lebesgue point of [MATH] |
[EQUATION] The lemma follows from these observations. Lemma 14 will be crucial to our proof that [MATH] is an embedding with probability [MATH] relative to |
[MATH] . In the case where [MATH] is fixed and only the observation function is perturbed, the proofs of injectivity and immersivity consider the ball [MATH] all at once. Such a thing is not possible here. Instead, we have to pick |
[MATH] satisfying [MATH] and localize around it and that is where Lemma 14 comes in. In order to localize around [MATH] , we adopt new notation that is centered at [MATH] . The re-centered diffeomorphism |
[MATH] is denoted by [MATH] . Similarly, [MATH] denotes [MATH] When we localize around [MATH] [MATH] will denote [MATH] The fixed point [MATH] is denoted [MATH] The fixed point [MATH] is denoted |
[MATH] Convention about [MATH] updated: [MATH] are iterates of [MATH] under [MATH] . Similarly, [MATH] are iterates of [MATH] under [MATH] |
Convention about [MATH] updated: [MATH] and [MATH] are iterates of [MATH] under [MATH] Convention about [MATH] updated: we assume [MATH] |
and [MATH] are obtained by iterating [MATH] All the lemmas of the previous section continue to hold after re-centering. The delay vector [MATH] defined in the previous section will be denoted by [MATH] if [MATH] is replaced by [MATH] . Similarly, if [MATH] is replaced by |
[MATH] in the definition of [MATH] we will denote the re-centered delay vector by [MATH] We may write [EQUATION] with the definition of [MATH] being the same as that of [MATH] but with [MATH] replaced by [MATH] . Likewise, |
[EQUATION] with a similar alteration of the definition of [MATH] to get [MATH] Finally, we note that the centered analogue of [MATH] |
is [MATH] Proof of injectivity In this section, our purpose is to prove that [MATH] defined in section 5, is injective for [MATH] . The assumptions about [MATH] and [MATH] are carried forward from earlier sections, although the third assumption about [MATH] is not necessary in its entirety. Further assumptions will be ... |
[MATH] in [MATH] for [MATH] Let us define [MATH] to be the set of [MATH] satisfying 1. [MATH] 2. [MATH] for [MATH] and [MATH] with [MATH] for each [MATH] |
In this section and the next, we always assume [MATH] Lemma 15 If [MATH] , every point of [MATH] is a Lebesgue point of [MATH] and therefore the probability of |
[MATH] relative to the open ball [MATH] is [MATH] Proof. Pick [MATH] satisfying [MATH] We will use an argument centered at [MATH] to show that |
[MATH] is a Lebesgue point of [MATH] Pick [MATH] so small that [MATH] for [MATH] . Define [MATH] as the set of [MATH] such that [MATH] |
for each [MATH] Let us look at [MATH] Using Lemma 13 and the definition of [MATH] , we get [EQUATION] with [MATH] and [EQUATION] |
There are two cases here. Suppose [MATH] is nonzero. Then by Lemma 11 (b), the rank of [MATH] is equal to the number of its rows. Therefore, the rank [MATH] is [MATH] . If in fact the corner entry [MATH] |
is zero, we can drop the last column and first row of [MATH] and conclude that the rank of [MATH] is [MATH] . In either case, the rank of [MATH] is [MATH] or greater. |
Define [MATH] , where the minimum is over [MATH] and [MATH] Cover [MATH] with [MATH] [MATH] -balls. Assumption about [MATH] (3): In ( 7.1 ), the [MATH] |
term is upper bounded by [MATH] . Like the earlier assumptions about [MATH] , this assumption too is a direct consequence of compactness. The earlier assumptions used [MATH] as a bound on Lipshitz constants. Here [MATH] is used as a bound on the Taylor series remainder. |
Now suppose [MATH] for some [MATH] and some [MATH] Because the Lipshitz constant of [MATH] with respect to [MATH] is bounded by [MATH] , we must have [MATH] |
at an [MATH] that is at the center of one the balls covering [MATH] Applying the nonlinear transfer of volume Lemma with [MATH] and [MATH] we find that the probability of [MATH] |
relative to [MATH] is upper bounded by [EQUATION] Because the number of fixed points is [MATH] and the number balls covering [MATH] is [MATH] , the probability of [MATH] for some |
[MATH] and some [MATH] relative to [MATH] is upper bounded by [EQUATION] Evidently, the probability goes to zero as [MATH] if [MATH] . Thus, we have shown that [MATH] is a Lebesgue point of [MATH] proving the lemma. |
Now define [MATH] to be the set of [MATH] satisfying 1. [MATH] 2. [MATH] for [MATH] with [MATH] for each [MATH] Lemma 16 If [MATH] , every point of [MATH] is a Lebesgue point of [MATH] and therefore the probability of |
[MATH] relative to [MATH] is [MATH] Proof. As before, we pick [MATH] satisfying [MATH] and will give an argument centered at [MATH] to show that [MATH] is a Lebesgue point of [MATH] As before, pick [MATH] so small that [MATH] |
for [MATH] . As before, define [MATH] as the set of [MATH] such that [MATH] for each [MATH] Using ( 5.2 ), we get [EQUATION] with [MATH] and |
[EQUATION] By Lemma 11 (c), the rank of [MATH] is equal to the number of its rows. Therefore, the rank of [MATH] is equal to [MATH] |
Define [MATH] , where the minimum is over [MATH] and [MATH] Cover [MATH] with [MATH] [MATH] -balls. Assumption about [MATH] (4): In ( 7.2 ), the [MATH] |
term is upper bounded by [MATH] . The first two assumptions about [MATH] are both obtained from upper bounds on the derivative of [MATH] or [MATH] with respect to [MATH] . This assumption as well as the preceding one are obtained from upper bounds on the second derivative. In all cases, the assumptions are direct conse... |
If [MATH] for some [MATH] , we must have [MATH] for some [MATH] that is the center of one of the balls covering [MATH] . Using the nonlinear transfer of volume Lemma , we find the probability of [MATH] for some |
[MATH] relative to the ball [MATH] to be upper bounded by [EQUATION] The limit of this probability as [MATH] is zero. It follows that [MATH] is a Lebesgue point of [MATH] |
completing the proof of this lemma. Lemma 16 allows us to conclude that the delay vectors of [MATH] and [MATH] do not coincide typically if [MATH] is a little removed from the fixed points of |
[MATH] . More generally, we need to argue that the delay vectors of [MATH] and [MATH] do not coincide for [MATH] . To make that argument, we define [MATH] |
to be the set of [MATH] satisfying 1. [MATH] 2. [MATH] for [MATH] with [MATH] for each [MATH] for [MATH] Lemma 17 For [MATH] and [MATH] , every point of [MATH] |
is a Lebesgue point of [MATH] and therefore the probability of [MATH] relative to the ball [MATH] is [MATH] Proof. The proof is almost identical to that of the previous lemma, which is a special case. The only significant difference occurs in the definition of [MATH] . In the general case, |
[EQUATION] Note that Lemma 11 (c) still applies, implying the rank of [MATH] to be equal to the number of its rows, because [MATH] |
The final lemma of this section pertains to the set [MATH] It is defined as the set of all [MATH] such that [MATH] and [MATH] provided |
1. [MATH] 2. [MATH] (which excludes the diagonal of [MATH] 3. [MATH] and [MATH] for [MATH] (so that both [MATH] and [MATH] stay away from fixed points) |
4. [MATH] and [MATH] for [MATH] (so that [MATH] does not come too close to the iterates of [MATH] and vice versa). Lemma 18 For [MATH] , every point of [MATH] is a Lebesgue point of [MATH] and therefore the probability of [MATH] relative to the ball [MATH] |
is [MATH] Proof. Again the argument begins by centering at some [MATH] satisfying [MATH] . However, the conditions on [MATH] this time are different. The radius [MATH] must be so small that for [MATH] the following conditions are satisfied: |
1. [MATH] 2. For any [MATH] [MATH] for [MATH] The set [MATH] is defined as the set of [MATH] satisfying the following conditions: |
1. [MATH] and [MATH] for [MATH] 2. [MATH] 3. [MATH] and [MATH] for [MATH] We have [EQUATION] The top row of [MATH] is zero. The rest of the [MATH] rows below are given by [MATH] |
[EQUATION] By Lemma 11 , the rank of [MATH] is equal to the number of its rows. Therefore the ranks of [MATH] and [MATH] are both equal to [MATH] |
Define [MATH] , where the minimum is over [MATH] and [MATH] Cover [MATH] with [MATH] balls. Assumption about [MATH] (5): The [MATH] |
term in ( 7.3 ) is upper bounded by [MATH] Suppose [MATH] for some [MATH] Then we must have [MATH] for an [MATH] that is at the center of one of the balls covering [MATH] . Applying the nonlinear transfer of volume Lemma , we find the probability of [MATH] for some |
[MATH] relative to the ball [MATH] to be upper bounded by [EQUATION] If [MATH] , the limit of this probability as [MATH] is [MATH] . Therefore, every [MATH] satisfying [MATH] |
is a Lebesgue point of [MATH] , which completes the proof of the lemma. We are now prepared to state and prove the main theorem of this section. |
Theorem 19 Assuming [MATH] and [MATH] satisfy the conditions laid down in section 6 and [MATH] , the delay mapping [MATH] is injective on the set [MATH] with probability one relative to the ball [MATH] |
Proof. The proof follows from Lemmas 15 17 and 18 by taking the limit [MATH] through a countable sequence. Proof of immersivity All the main techniques have been demonstrated in the proof of injectivity of the delay mapping [MATH] . The assumption in section 6 that |
[MATH] is immersive at all fixed points in [MATH] simplifies the proof of immersivity considerably. Define [MATH] as the set of all [MATH] satisfying |
[MATH] and [MATH] is immersive at all [MATH] satisfying [MATH] for [MATH] In other words, we are requiring [MATH] if [MATH] and [MATH] is removed from each periodic point by at least [MATH] |
Lemma 20 For [MATH] , every point of [MATH] is a Lebesgue point of [MATH] and therefore the probability of [MATH] relative to [MATH] is |
[MATH] Proof. We center at [MATH] satisfying [MATH] as before. Again as before, we assume [MATH] to be so small that [MATH] for [MATH] |
Define [MATH] to be the set of all [MATH] satisfying [MATH] for [MATH] Then [EQUATION] with [EQUATION] By Lemma 12 , the rank of [MATH] is |
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