Consider a continuous map T: 𝐑^n →𝐑^n such that 𝐓 ∈𝐂^1 and 𝐓(0)=0. Set A={x |T(x)<x}, B={x |T(x)

Step 1: u000AWe start by analyzing the sets u005C( A u005C) and u005C( B u005C). The set u005C( A u005C) consists of points

Consider a continuous map T: 𝐑^n →𝐑^n such that 𝐓 ∈𝐂^1 and...

Consider a continuous map $T: \mathbf{R}^n \rightarrow \mathbf{R}^n$ such that $\mathbf{T} \in \mathbf{C}^1$ and $\mathbf{T}(0)=0$. Set $A=\{x \mid\|T(x)\|<\|x\|\}, B=\{x \mid\|T(x)\| \geq\|x\|\}$. Assume that (a) A is invariant under $T, T(A) \subseteq A$; (b) For all $b \in B$, there exists a positive integer $i(b)$ such that $T^{(b)}(b) \in A$. Show that for $x \in R^n, T^k(x) \rightarrow 0$ as $k \rightarrow \infty$.

Consider a continuous map $T: \mathbf{R}^n \rightarrow \mathbf{R}^n$ such that $\mathbf{T} \in \mathbf{C}^1$ and $\mathbf{T}(0)=0$. Set $A=\{x \mid\|T(x)\|<\|x\|\}, B=\{x \mid\|T(x)\| \geq\|x\|\}$. Assume that
(a) A is invariant under $T, T(A) \subseteq A$;
(b) For all $b \in B$, there exists a positive integer $i(b)$ such that $T^{(b)}(b) \in A$. Show that for $x \in R^n, T^k(x) \rightarrow 0$ as $k \rightarrow \infty$.

Key Concepts

Fixed Points and Stability

A fixed point is a point that is mapped to itself by a function, and its stability refers to the behavior of orbits under perturbations of this point. Since T(0) = 0 is a fixed point and all trajectories eventually decrease in norm (once they enter A), the origin acts as an attracting fixed point. Its attracting nature implies that regardless of the starting position in ??, the iterated application of T will eventually bring points arbitrarily close to 0, establishing stability of the fixed point at the origin.

Contractive Behavior and Monotonic Decrease

Contractive behavior describes a situation where the distance between points decreases under repeated application of a map. In the given problem, once a point’s orbit enters the set A, the norm of its image is strictly less than that of the point itself, ensuring a monotonic decrease. This property, which may be analyzed using ideas similar to a contraction mapping (even if only locally valid within A), is central in arguing that the iterates will eventually converge to the origin.

Basin of Attraction

The basin of attraction of a fixed point is the set of all initial conditions whose trajectories eventually converge to that fixed point. Here, the fixed point is the origin. The conditions of the problem ensure that every point in ?? eventually enters the set A, within which the norm of the map’s iterates is strictly decreasing. This guarantees that all points, regardless of where they start, are attracted to 0 as the number of iterations increases.

Invariant Set

An invariant set is a subset of the phase space which, under the application of a map or a flow, is mapped into itself. In dynamical systems, identifying an invariant set helps in understanding the behavior of trajectories that begin in that set because they remain there for all time. In this context, the set A is invariant under the map T, meaning that if a point starts in A, all its forward iterates will remain in A, which plays a crucial role in establishing the eventual decrease in the norm of the trajectories contained in A.

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