Attraction rate and behavior of the orbits.

Question

Let $X$ be a Hilbert space and $T:X\to X$ be a $C^1$-differentiable map. Assume that $K\subset X$ is a compact subset with $TK=K$ and $U\subset X$ is a neighborhood of $K$ such that $TU\subset U$. Suppose that $K$ attracts $U$, that is, there exists $\gamma\in (0,1)$ and $C>0$ such that $$dist_H(T^nU,K):=\sup\limits_{x\in T^nU}d(x,K)\le C\gamma^n, \ \forall n\in\mathbb{N},$$ where $d(x,K)=\inf\limits_{k\in K}\parallel x-k\parallel$.

I want to know if there exists $\lambda\in (0,1)$ and $N\in\mathbb{N}$ such that $$d(T^Nx,K)\le \lambda d(x,K),\ \forall x\in U.$$

Briefly, I want to know if exponential attraction guarantees that the points are always getting closer to $K$ when we applie the operator $T^N$. If its not true, can we put conditions on the metric space $X$ and $T$ such that this property holds?

Thank you so much for your help! Any idea will be appreciated.

Answer 1

Let $D_xT$ be derivative of $T$ at $x \in U$.

Then for all $x \in T^{n-1}U$, $d(Tx,K) \leq C\gamma^n \implies \inf_{k \in K} ||Tx-k|| \leq C\gamma^n \implies ||Tx-Tk_x|| \leq C\gamma^n$.

We now have, $Tx-Tk_x = D_xT(x-k_x) + O(||x-k_x||^2)$

$||Tx-Tk_x|| = ||D_xT(x-k_x)|| + O(||x-k_x||^2) \leq C \gamma^n \implies ||D_xT(x-k_x)|| + O(||x-k_x||^2) \leq C \gamma^n \implies ||D_xT(x-k_x)|| \leq C \gamma^n + O(||x-k_x||^2) $

We also have $||D_xT(x-k_x)|| \leq ||D_xT|| \times ||x-k_x||$.

An educated guess from above would be $||D_yT(x-k)|| \leq \gamma \times ||x-k||$ for all $k \in \partial K$ such that $y=k+\lambda(x-k)$ for all sufficiently small $\lambda$ and for all $x$ in a small neighbourhood of $k$.

Lets take $k \in \partial K$ and such that $||D_yT(x-k)|| > \gamma \times ||x-k||$ for a particular $x \in U_r:=B(k,r)$ and for all $\lambda \leq \lambda^*$.

Also we have by mean value theorem for some $y = k+\lambda(x-k)$, $Tx-Tk = D_{y}T(x-k)$.

If we assume, $K=\{0\}$, $X=\mathbb{R}$, we have from above, $|Tx| > \gamma \times |x| $ for all $x \in [-r/2,r/2]$. Hence we have $TU_r$ will not shrink at rate less than or equal to $\gamma$. A contradiction when at some point $T^nU \subseteq U_r$. Hence when $K=\{0\}$, $X=\mathbb{R}$, A necessary condition is $|T'(z)| \leq \gamma$ for all $z \in [-r/2,r/2]$ for some $r>0$.Hence $U_r$ also shrinks at same rate $\gamma$. Hence we can have $\lambda = C \gamma^p <1$ with $N=p$.

I think this can be extended to any $X=\mathbb{R}^m$ for $K = \{k\}$. The key is the mean value theorem. See if you can generalize this to other $X$.