Shadowing near nonhyperbolic fixed points

We use Lyapunov type functions to find conditions of finite shadowing in a neighborhood of a nonhyperbolic fixed point of a one-dimensional or two-dimensional homeomorphism or diffeomorphism. A new concept of shadowing in which we control the size of one-step errors is introduced in the case of a nonisolated fixed point.

In the study of shadowing in noninvariant sets (such as neighborhoods of fixed points), the concept of finite shadowing is natural.
We say that f has the finite shadowing property in a set K ⊂ M if, for any ε > 0, there exists a d > 0 such that if {p k ∈ K : 0 ≤ k ≤ m} is a d-pseudotrajectory of f , then there exists a point r such that Let us emphasize that in the above definition, d depends on K and ε but not on m.
The structure of the paper is as follows. In Sec. 2, we treat the one-dimensional case. We prove a simple general statement (Theorem 2.1) and show that, in some cases, the dependence of d on ε can be clarified. Section 3 is devoted to the method of Lyapunov type functions developed in [7]. In Sec. 4, we give general conditions of finite shadowing in the two-dimensional case and then treat in detail an important example of a diffeomorphism of the form f (x, y) = (x − x 2n+1 + X(x, y), y + y 2m+1 + Y (x, y)), where n, m are natural numbers and X, Y are smooth functions that vanish at the origin together with their Jacobi matrices. Finally, Sec. 5 is devoted to shadowing near a nonisolated fixed point. We study a simple (but nontrivial) example of the diffeomorphism for which the origin is a nonisolated fixed point (any point (0, y) is a fixed one).
Of course, such a system does not have the usual shadowing property. In this case, we work with a new concept of shadowing in which we control the size of one-step errors.
Our methods can be applied to dynamical systems with phase space of arbitrary dimension; in this paper, we restrict the consideration to one-dimensional and twodimensional systems to clarify the presentation of the main ideas.
2. One-dimensional case. First we consider the problem of shadowing near a nonhyperbolic fixed point of a homeomorphism in the simplest, one-dimensional, case.
Let f be a homeomorphism of a neighborhood U of a fixed point 0 ∈ R to its image.
We consider in detail the case where f is nonhyperbolically expanding in a neighborhood of a fixed point; the case of nonhyperbolic contraction near a fixed point is treated similarly.
We impose simplest possible conditions on f ; in a sense, precisely this topological form of conditions is generalized by our conditions in the two-dimensional case. Condition 1. There exist numbers a, A > 0 such that if |x| ≤ A and 0 < v < a, then f Denote by B(r, A) the closed r-neighborhood of a set A. Proof. Fix ε > 0; we assume that ε ≤ a.
The compactness of the set B and the continuity of f imply that there exists a d > 0 such that Similarly one shows that y > s, which proves (6).
It follows from inclusions (6) that the set is not empty. Clearly, for any point r ∈ C, inequalities (1) hold.
In Theorem 2.1, we can say nothing about the dependence of d on ε. For a particular example of a diffeomorphism with a nonhyperbolic fixed point considered below, such a dependence can be clarified.
where n is a natural number. Take ε > 0 and let Then S(x, ε) is a polynomial in x of even degree with positive leading coefficient. Since the derivative of S(x, ε) in x has a unique zero x = −ε/2, the inequality holds. Thus, the form S(x, ε) − ε 2n 1 + 2 2n of degree 2n is positive definite, and there exists a positive number α = α(n) such that Assume that Then there exist A, a > 0 such that if |x| ≤ A and 0 < ε < a, then Let If {p k } is a d-pseudotrajectory with |p k | ≤ A, then it follows from (7) and (9) that This relation and a similar relation for f (p k − ε) − f (p k ) mean that an analog of inclusion (5) holds for any p k .
Repeating the proof of Theorem 1, we conclude that f has the finite shadowing property in the set B(A, 0). Note that, for example, condition Our reasoning also shows that if X(x) ≡ 0, then f has the finite shadowing property in the whole line R with the same dependence of d on ε given by (10).

Remark 1.
In [8], S. Tikhomirov used a different approach to show that the diffeomorphism f (x) = x + x 3 has the shadowing property with d = cε 3 .
3. Lyapunov functions and shadowing. As was mentioned in the Introduction, we consider in detail the problem of finite shadowing for a homeomorphism f of the plane R 2 in two cases: Case I (the origin is an isolated nonhyperbolic fixed point) and Case NI (the y-axis consists of fixed points).
We apply the approach based on pairs of Lyapunov type functions developed in the paper [7].
Let us formulate the sufficient conditions of finite shadowing obtained in [7] in a form modified to fit our purposes in this paper.
Let K 0 = R 2 in Case I, and let in Case NI. We assume that there exist two continuous nonnegative functions V and W defined on K 0 × K 0 such that V (p, p) = W (p, p) = 0 for any p ∈ K 0 and the conditions (C1)-(C9) stated below are satisfied.
We formulate our conditions not directly in terms of the functions W and V but in terms of some geometric objects defined via these functions.
Fix a positive number δ and a point p ∈ K 0 and let Int 0 Q(δ, p) = {q ∈ P (δ, p) : V (q, p) = δ, W (q, p) < δ}. Let K be a subset of K 0 (in our basic examples, K is a small closed neigborhood of the origin in Case I and K is a neighborhood of the origin in K 0 in Case NI).
(C1) For any ε > 0 there exists In the next group of conditions, we state our assumptions on the behavior of the introduced objects and their images under the homeomorphism f .
Let p, q ∈ K and let 0 < δ < ∆. We say that condition Here P = P (δ, p), The same reasoning as in [7] proves the following statement.
Thus, to show that f has the finite shadowing property in a neighborhood K of the origin, it is enough to find functions V and W that satisfy conditions (C1) -(C4) and to show that for any ∆ > 0 there exists a δ ∈ (0, ∆) with the following property: There exists a d > 0 such that if p, q ∈ K and |q − f (p)| ≤ d, then condition W(δ, ∆, p, q) is satisfied.
Indeed, take any ε > 0, find a corresponding ∆ 0 (see condition (C1)), then take suitable ∆ < ∆ 0 and δ, and finally find a d > 0 having the above property. Then, if p 0 , . . . , p m ∈ K is a d-pseudotrajectory of f , this pseudotrajectory is ε-shadowed by any point r that satisfies inclusions (11).
We realize this scheme in the next section considering Case I. In Case NI, f does not have the usual shadowing property, and we have to modify the concept of shadowing (see Sec. 5).
4. Two-dimensional case. Isolated fixed point. Now we consider a twodimensional homeomorphism f (x, y) = (g(x, y), h(x, y)) having a fixed point at the origin and assume that f is contracting in the direction of variable x and expanding in the direction of variable y (and both the contraction and expansion are not assumed to be hyperbolic).
Let p = (p x , p y ) be the coordinate representation of a point p ∈ R 2 . In the case considered, we introduce two functions, For such functions, conditions (C1) -(C4) are obviously satisfied for any p ∈ R 2 and any 0 < δ < ∆.
We first formulate general conditions of finite shadowing for an arbitrary compact subset K of the plane and then apply them to our first basic example.

Condition 2.
For any ∆ 0 > 0 there exist δ, ∆ > 0 such that δ < ∆ < ∆ 0 and if p ∈ K, then condition (C5) with q = f (p) is satisfied, where and Theorem 4.1. If K is a compact subset of the plane and condition 2 is satisfied, then f has the finite shadowing property in the set K Proof. First we show that condition 2 implies that condition W(δ, ∆, p, f (p)) is satisfied for any p ∈ K.
Finally, we note that since K is compact and f , V , and W are continuous, the form of conditions (C5) -(C9) implies that there exists a d > 0 depending only on δ and ∆ such that if p ∈ K and |q − f (p)| < d, then condition W(δ, ∆, p, q) is satisfied. Now Theorem 4.1 is a corollary of the proposition stated in the previous section.
Example 2. Consider a diffeomorphism (2) in which n, m are natural numbers and X, Y are smooth functions that vanish at the origin together with their Jacobi matrices.
First we fix a small closed neighborhood K of the origin (in what follows, we make it as small as our future conditions require).
Let us assume that if p ∈ K, then and We take the same functions V and W as above in this section. It follows from the form of f that for any α > 0 we can find a neighborhood K such that ∂f ∂(x, y) Thus, if α > 0 is given, then, for δ small enough (and K properly chosen), condition (C5) is satisfied with ∆ = (1 + α)δ.
We take α < 1 (in which case we may take ∆ = 2δ in condition (C5)) and assume that K is so small that ∂Y ∂x (p) ≤ 1 4 , p ∈ K.
Assume, for definiteness, that w > 0 and estimate, using condition (17): where we take into account that p y + t is not identically zero. This proves condition (14) (the case w < 0 is treated similarly). To prove condition (12), we consider the case where |v| = δ, |w| ≤ d; the case where w = 0 is treated similarly (note that in both cases, w = νv with |ν| ≤ 1).
Let us consider as a "test perturbation" a monomial Y (x, y) = ax k y l in (2), where a ∈ R, k ≥ 0, and l ≥ 1. Let (x, y) be a point in a neighborhood K of the origin with y = 0.
Taking ν = 0 in (17) and dividing the result by y 2m , we get the following necessary condition: This condition is obviously satisfied in a small K if l ≥ 2m + 1 (and if |a| is small in the case where l = 2m + 1 and k = 0). If l < 2m + 1, then the necessary condition looks as follows: a > 0, k is even, and l is odd.
If l ≥ 2m + 1 (and if |a| is small in the case where l = 2m + 1 and k = 0), condition (17) is obviously satisfied in a small neighborhood of the origin.
In the case where l ≤ 2m, we get one more necessary condition: k + l ≥ 2m + 1. Indeed, since a > 0, k is even, and l is odd, it is enough to consider (17) for x, y ≥ 0. Now let us write (17) in the form ax k−1 y l−1 (ky − lx) < (2m + 1)y 2m .
If k + l < 2m + 1, set x = kb 2l and y = b with small b > 0. We get an inequality of the form 0 < const < b 2m−k−l+1 which cannot be satisfied for all small b.
Elementary calculations show that if l ≤ 2m, a > 0, k is even, l is odd, and k + l ≥ 2m + 1, then condition (19) is satisfied in a small neighborhood of the origin.
We can obtain similar conditions if X(x, y) in (2) is also a monomial.
Our methods allow us to estimate the dependence of d on ε in the finite shadowing property for the considered case.
For example, if X(x, y) = a 1 x k1 y l1 and Y (x, y) = a 2 x k2 y l2 in (2) with k 1 > 2n+1 and l 2 > 2m + 1, then the same reasoning as above shows that there exists a neighborhood K of the origin and a small number c > 0 such that if a δ > 0 is given and {p k } is a finite set of points in K with where p = max(2n+1, 2m+1), then conditions W(δ, 2δ, p k , p k+1 ) are satisfied. This means that f has in K the finite shadowing property with the following dependence of d on ε: d = cε p . 5. Two-dimensional case. Nonisolated fixed point. In this section, we consider a model example of the diffeomorphism (3) for which the origin is a nonisolated fixed point (any point (0, y) is a fixed one).
Of course, f does not have the shadowing property. Nevertheless, we show that f has an analog of the finite shadowing property if we consider pseudotrajectories {p k } with (p k ) x = 0 and allow the "errors" |f (p k ) − p k+1 | to depend on (p k ) x . The errors must be smaller for smaller values of |(p k ) x |. Such an approach (in the case of a nontransverse homoclinic point) has been suggested by S. Tikhomirov.
We restrict our consideration to the case of a diffeomorphism f of a very simple form (3) to simplify presentation (as the reader will see, even this case is not completely trivial); of course, our reasoning can be applied in more general situations.
Note that Thus, we consider a finite pseudotrajectory p 0 , . . . , p m and assume that (p k ) x = 0 and for some d > 0. Our main result is as follows. Recall that in our case, Theorem 5.1. There exists a neighborhood K of the origin and a number c > 0 such that, for any ε > 0 and any pseudotrajectory p 0 , . . . , p m in K ∩K 0 that satisfies conditions (21) with d = cε there exists a point r for which inequalities (1) are satisfied.
Proof. As above, in our proof we use the approach based on Lyapunov functions, but now one of the functions is modified. We take .
Clearly, these functions are nonnegative and continuous on K 0 ×K 0 and they vanish if their arguments coincide. It is obvious that conditions (C1) -(C4) are satisfied (and we can take ∆ 0 (ε) = ε/2 in condition (C1)).
In the following proof, we take (the constant N is chosen below) and d = cδ in condition (21). First we take c = 1 and then make c smaller preserving the same notation c; the same is done with the neighborhood K.
Our main goal is to check condition W(δ, ∆, p k , p k+1 ) for consecutive points p k , p k+1 of the pseudotrajectory considered for properly chosen δ and ∆; after that, we apply Proposition 1 stated in Sec. 3.
To make the presentation shorter, let p k = p = (x, y) and p k+1 = q = (x , y ). Thus, we may assume that |x|, |x |, |y|, |y | are as small as we need. First we claim that there exists a number N > 0 such that if K is a small neigborhood of the origin, p, q ∈ K, and δ < 1, then inclusions (C5) hold with ∆ = N δ.
Let us prove our statement on the existence of a number N for which the third of the above inequalities holds (the remaining inequalities are established using a similar reasoning).
We note that Thus, the projection of f (T (δ, p)) to the x axis is the segment At the same time, if x = x/2 + u, then the projection of P (δ, q) to the x axis is the and |u| ≤ dx 2 (in the above formulas, we note that if d and x are small, then x + 2u > 0). Since it is easy to understand that there exists a c > 0 (independent of x and δ) such that if d ≤ cδ in (21), then To complete the proof of condition (C6) and prove condition (C7), we note that If y = y(1 + x 2 ) + u, then the y coordinate of any point of the set Q(δ, q) is either y − δ or y + δ. Let us represent y + δ − y(1 + (x + v) 2 ) = δ − y(2xv + v 2 ) + u.
Let us represent

The estimates
δ(x + v) 2 ≥ δx 2 (1 − δ) 2 and |y(2xv + v 2 )| ≤ |y|(2δx 2 + δ 2 x 2 ) ≤ δx 2 /2 (which is valid if |y| and δ do not exceed a small value independent of x) imply that there exists a constant c (inependent of x, y, and δ) such that if u ≤ cδx 2 , then the projection of the point f (r) to the y axis does not belong to the segment D.
Applying a similar reasoning to points r = (x + v, y − δ), we complete the proof of condition (C9). Estimates of a similar form prove condition (C8).
To complete the proof of Theorem 5.1, we take into account the equality d = cδ and relations (22).