I am working on an alternative proof of Parabolic Implosion from Complex Dynamics, but only allowing hyperbolic perturbation

Theorem (Parabolic Implosion) Let $f(z)=z^2+z$ and $U_f$ be parabolic basin of $f$. Let $\mathcal{H}_2$ to be the main hyperbolic component (main cardioid). Then for all $z_0\in U_f$ and $\epsilon>0$, there exist $f_n \in \mathcal{H}_2$ arbitrarily close to $f$ so that $J(f_n) \cap B(z_0, \epsilon) \neq \emptyset$.

My strategy is to do contradiction. Assume the opposite, there exist $z_0 \in U_f$ and $\epsilon>0$ so for all $f_n \rightarrow f$ with $f_n\in \mathcal{H}_2$, we have $B(z_0, \epsilon) \subset U_{f_n}$ where $U_{f_n}$ is the hyperbolic basin of $f_n$.

Recall Caratheodory topology of Pointed Disk, roughly says $(U_n, z_n) \rightarrow (U_\infty, z_\infty)$ iff their Riemann map converges.

A classic fact is the set of pointed disks $(U, 0)$ containing $B(0, r)$ for some $r > 0$ is compact in Caratheodory.

The fact $B(z_0, \epsilon) \subset U_{f_n}$ means we can always extract a subsequential convergence out of $(U_{f_n}, z_0)$ pointed disk in Caratheodory topology.

Therefore I have $(U_{f_n}, z_0) \rightarrow (U_{\infty,z_0}, z_0)$ in Caratheodory. After some argument, you only have to deal with $U_{\infty,z_0} \subset U_f$.

Now $U_{\infty,z_0}$ really depends on $z_0$ and sequence $f_n$. Different basepoint can lead to different Caratheodory limit. That is, given $z_0 \neq z_1$, we might have $U_{\infty, z_0} \neq U_{\infty, z_1}$. Similarly different sequence will give different limit. In my context, lets fix a sequence, so we only worry about base point.

Question: I want to say for all $z_0$, there exist $k>0$ so $f^k(U_{\infty,z_0}) = U_{\infty,z_0}$ i.e. eventually forward invariant domain. Somehow I want to quote No Wandering Domain theorem, but this $(U_{\infty, z_0}$ is not quite a domain but sub-domain. See picture below due to Arnaud Cheritat.

Here $(U_\infty,z_0)$ is the bottom left region, gets mapped to top right, which stays forward invariant

Now, forgetting Caratheodory topology and only use Hausdorff topology, we can say $U_{f_n} \rightarrow V$ in Hausdorff where $V$ open. We also can argue $V \subsetneq U_f$. Because $f_n(U_{f_n}) = U_{f_n}$, we immediately get $f(V)=V$, giving a sub-dynamical system of parabolic basin.

Perhaps easier to transfer the picture to $\mathbb{D}$ disk, so uniformize it and see we have a sub-dynamical system of a doubly parabolic Blaschke product. What are the possibility for such sub-dynamics?

I am working on an alternative proof of parabolic implosion from complex dynamics, but only allowing hyperbolic perturbation.

Theorem (Parabolic Implosion) Let $f(z)=z^2+z$ and $U_f$ be parabolic basin of $f$. Let $\mathcal{H}_2$ to be the main hyperbolic component (main cardioid). Then for all $z_0\in U_f$ and $\epsilon>0$, there exist $f_n \in \mathcal{H}_2$ arbitrarily close to $f$ so that $J(f_n) \cap B(z_0, \epsilon) \neq \emptyset$.

My strategy is to do contradiction. Assume the opposite, there exist $z_0 \in U_f$ and $\epsilon>0$ so for all $f_n \rightarrow f$ with $f_n\in \mathcal{H}_2$, we have $B(z_0, \epsilon) \subset U_{f_n}$ where $U_{f_n}$ is the hyperbolic basin of $f_n$.

Recall Caratheodory topology of Pointed Disk, roughly says $(U_n, z_n) \rightarrow (U_\infty, z_\infty)$ iff their Riemann map converges.

A classic fact is the set of pointed disks $(U, 0)$ containing $B(0, r)$ for some $r > 0$ is compact in Caratheodory.

The fact $B(z_0, \epsilon) \subset U_{f_n}$ means we can always extract a subsequential convergence out of $(U_{f_n}, z_0)$ pointed disk in Caratheodory topology.

Therefore I have $(U_{f_n}, z_0) \rightarrow (U_{\infty,z_0}, z_0)$ in Caratheodory. After some argument, you only have to deal with $U_{\infty,z_0} \subset U_f$.

Now $U_{\infty,z_0}$ really depends on $z_0$ and sequence $f_n$. Different basepoint can lead to different Caratheodory limit. That is, given $z_0 \neq z_1$, we might have $U_{\infty, z_0} \neq U_{\infty, z_1}$. Similarly different sequence will give different limit. In my context, lets fix a sequence, so we only worry about base point.

Question: I want to say for all $z_0$, there exist $k>0$ so $f^k(U_{\infty,z_0}) = U_{\infty,z_0}$ i.e. eventually forward invariant domain. Somehow I want to quote No Wandering Domain theorem, but this $(U_{\infty, z_0}$ is not quite a domain but sub-domain. See picture below due to Arnaud Cheritat.

Here $(U_\infty,z_0)$ is the bottom left region, gets mapped to top right, which stays forward invariant

Now, forgetting Caratheodory topology and only use Hausdorff topology, we can say $U_{f_n} \rightarrow V$ in Hausdorff where $V$ open. We also can argue $V \subsetneq U_f$. Because $f_n(U_{f_n}) = U_{f_n}$, we immediately get $f(V)=V$, giving a sub-dynamical system of parabolic basin.

Perhaps easier to transfer the picture to $\mathbb{D}$ disk, so uniformize it and see we have a sub-dynamical system of a doubly parabolic Blaschke product. What are the possibility for such sub-dynamics?

I am working on an alternative proof of Parabolic Implosion from Complex Dynamics, but only allowing hyperbolic perturbation

Theorem (Parabolic Implosion) Let $f(z)=z^2+z$ and $U_f$ be parabolic basin of $f$. Let $\mathcal{H}_2$ to be the main hyperbolic component (main cardioid). Then for all $z_0\in U_f$ and $\epsilon>0$, there exist $f_n \in \mathcal{H}_2$ arbitrarily close to $f$ so that $J(f_n) \cap B(z_0, \epsilon) \neq \emptyset$.

My strategy is to do contradiction. Assume the opposite, there exist $z_0 \in U_f$ and $\epsilon>0$ so for all $f_n \rightarrow f$ with $f_n\in \mathcal{H}_2$, we have $B(z_0, \epsilon) \subset U_{f_n}$ where $U_{f_n}$ is the hyperbolic basin of $f_n$.

Recall Caratheodory topology of Pointed Disk, roughly says $(U_n, z_n) \rightarrow (U_\infty, z_\infty)$ iff their Riemann map converges.

A classic fact is the set of pointed disks $(U, 0)$ containing $B(0, r)$ for some $r > 0$ is compact in Caratheodory.

The fact $B(z_0, \epsilon) \subset U_{f_n}$ means we can always extract a subsequential convergence out of $(U_{f_n}, z_0)$ pointed disk in Caratheodory topology.

Therefore I have $(U_{f_n}, z_0) \rightarrow (U_{\infty,z_0}, z_0)$ in Caratheodory. After some argument, you only have to deal with $U_{\infty,z_0} \subset U_f$.

Now $U_{\infty,z_0}$ really depends on $z_0$ and sequence $f_n$. Different basepoint can lead to different Caratheodory limit. That is, given $z_0 \neq z_1$, we might have $U_{\infty, z_0} \neq U_{\infty, z_1}$. Similarly different sequence will give different limit. In my context, lets fix a sequence, so we only worry about base point.

Question: I want to say for all $z_0$, there exist $k>0$ so $f^k(U_{\infty,z_0}) = U_{\infty,z_0}$ i.e. eventually forward invariant domain. See picture below due to Arnaud Cheritat.

Here $(U_\infty,z_0)$ is the bottom left region, gets mapped to top right, which stays forward invariant

Now, forgetting Caratheodory topology and only use Hausdorff topology, we can say $U_{f_n} \rightarrow V$ in Hausdorff where $V$ open. We also can argue $V \subsetneq U_f$. Because $f_n(U_{f_n}) = U_{f_n}$, we immediately get $f(V)=V$, giving a sub-dynamical system of parabolic basin.

Perhaps easier to transfer the picture to $\mathbb{D}$ disk, so uniformize it and see we have a sub-dynamical system of a doubly parabolic Blaschke product. What are the possibility for such sub-dynamics?

I am working on an alternative proof of Parabolic Implosion from Complex Dynamics, but only allowing hyperbolic perturbation

Theorem (Parabolic Implosion) Let $f(z)=z^2+z$ and $U_f$ be parabolic basin of $f$. Let $\mathcal{H}_2$ to be the main hyperbolic component (main cardioid). Then for all $z_0\in U_f$ and $\epsilon>0$, there exist $f_n \in \mathcal{H}_2$ arbitrarily close to $f$ so that $J(f_n) \cap B(z_0, \epsilon) \neq \emptyset$.

My strategy is to do contradiction. Assume the opposite, there exist $z_0 \in U_f$ and $\epsilon>0$ so for all $f_n \rightarrow f$ with $f_n\in \mathcal{H}_2$, we have $B(z_0, \epsilon) \subset U_{f_n}$ where $U_{f_n}$ is the hyperbolic basin of $f_n$.

Recall Caratheodory topology of Pointed Disk, roughly says $(U_n, z_n) \rightarrow (U_\infty, z_\infty)$ iff their Riemann map converges.

A classic fact is the set of pointed disks $(U, 0)$ containing $B(0, r)$ for some $r > 0$ is compact in Caratheodory.

The fact $B(z_0, \epsilon) \subset U_{f_n}$ means we can always extract a subsequential convergence out of $(U_{f_n}, z_0)$ pointed disk in Caratheodory topology.

Therefore I have $(U_{f_n}, z_0) \rightarrow (U_{\infty,z_0}, z_0)$ in Caratheodory. After some argument, you only have to deal with $U_{\infty,z_0} \subset U_f$.

Now $U_{\infty,z_0}$ really depends on $z_0$ and sequence $f_n$. Different basepoint can lead to different Caratheodory limit. That is, given $z_0 \neq z_1$, we might have $U_{\infty, z_0} \neq U_{\infty, z_1}$. Similarly different sequence will give different limit. In my context, lets fix a sequence, so we only worry about base point.

Question: I want to say for all $z_0$, there exist $k>0$ so $f^k(U_{\infty,z_0}) = U_{\infty,z_0}$ i.e. eventually forward invariant domain. Somehow I want to quote No Wandering Domain theorem, but this $(U_{\infty, z_0}$ is not quite a domain but sub-domain. See picture below due to Arnaud Cheritat.

Here $(U_\infty,z_0)$ is the bottom left region, gets mapped to top right, which stays forward invariant

Now, forgetting Caratheodory topology and only use Hausdorff topology, we can say $U_{f_n} \rightarrow V$ in Hausdorff where $V$ open. We also can argue $V \subsetneq U_f$. Because $f_n(U_{f_n}) = U_{f_n}$, we immediately get $f(V)=V$, giving a sub-dynamical system of parabolic basin.

Perhaps easier to transfer the picture to $\mathbb{D}$ disk, so uniformize it and see we have a sub-dynamical system of a doubly parabolic Blaschke product. What are the possibility for such sub-dynamics?

I am working on an alternative proof of Parabolic Implosion from Complex Dynamics, but only allowing hyperbolic perturbation

Theorem (Parabolic Implosion) Let $f(z)=z^2+z$ and $U_f$ be parabolic basin of $f$. Let $\mathcal{H}_2$ to be the main hyperbolic component (main cardioid). Then for all $z_0\in U_f$ and $\epsilon>0$, there exist $f_n \in \mathcal{H}_2$ arbitrarily close to $f$ so that $J(f_n) \cap B(z_0, \epsilon) \neq \emptyset$.

My strategy is to do contradiction. Assume the opposite, there exist $z_0 \in U_f$ and $\epsilon>0$ so for all $f_n \rightarrow f$ with $f_n\in \mathcal{H}_2$, we have $B(z_0, \epsilon) \subset U_{f_n}$ where $U_{f_n}$ is the hyperbolic basin of $f_n$.

Recall Caratheodory topology of Pointed Disk, roughly says $(U_n, z_n) \rightarrow (U_\infty, z_\infty)$ iff their Riemann map converges.

A classic fact is the set of pointed disks $(U, 0)$ containing $B(0, r)$ for some $r > 0$ is compact in Caratheodory.

The fact $B(z_0, \epsilon) \subset U_{f_n}$ means we can always extract a subsequential convergence out of $(U_{f_n}, z_0)$ pointed disk in Caratheodory topology.

Therefore I have $(U_{f_n}, z_0) \rightarrow (U_{\infty,z_0}, z_0)$ in Caratheodory. After some argument, you only have to deal with $U_\infty \subset U_f$.

Now $U_{\infty,z_0}$ really depends on $z_0$ and sequence $f_n$. Different basepoint can lead to different Caratheodory limit. That is, given $z_0 \neq z_1$, we might have $U_{\infty, z_0} \neq U_{\infty, z_1}$. Similarly different sequence will give different limit. In my context, lets fix a sequence, so we only worry about base point.

Question: I want to say for all $z_0$, there exist $k>0$ so $f^k(U_{\infty,z_0}) = U_{\infty,z_0}$ i.e. eventually periodic. See picture below due to Arnaud Cheritat.

Here $(U_\infty,z_0)$ is the bottom left region, gets mapped to top right, which stays forward invariant

Now, forgetting Caratheodory topology and only use Hausdorff topology, we can say $U_{f_n} \rightarrow V$ in Hausdorff where $V$ open. We also can argue $V \subsetneq U_f$. Because $f_n(U_{f_n}) = U_{f_n}$, we immediately get $f(V)=V$, giving a sub-dynamical system of parabolic basin.

Perhaps easier to transfer the picture to $\mathbb{D}$ disk, so uniformize it and see we have a sub-dynamical system of a doubly parabolic Blaschke product. What are the possibility for such sub-dynamics?

I am working on an alternative proof of Parabolic Implosion from Complex Dynamics, but only allowing hyperbolic perturbation

Theorem (Parabolic Implosion) Let $f(z)=z^2+z$ and $U_f$ be parabolic basin of $f$. Let $\mathcal{H}_2$ to be the main hyperbolic component (main cardioid). Then for all $z_0\in U_f$ and $\epsilon>0$, there exist $f_n \in \mathcal{H}_2$ arbitrarily close to $f$ so that $J(f_n) \cap B(z_0, \epsilon) \neq \emptyset$.

My strategy is to do contradiction. Assume the opposite, there exist $z_0 \in U_f$ and $\epsilon>0$ so for all $f_n \rightarrow f$ with $f_n\in \mathcal{H}_2$, we have $B(z_0, \epsilon) \subset U_{f_n}$ where $U_{f_n}$ is the hyperbolic basin of $f_n$.

Recall Caratheodory topology of Pointed Disk, roughly says $(U_n, z_n) \rightarrow (U_\infty, z_\infty)$ iff their Riemann map converges.

A classic fact is the set of pointed disks $(U, 0)$ containing $B(0, r)$ for some $r > 0$ is compact in Caratheodory.

The fact $B(z_0, \epsilon) \subset U_{f_n}$ means we can always extract a subsequential convergence out of $(U_{f_n}, z_0)$ pointed disk in Caratheodory topology.

Therefore I have $(U_{f_n}, z_0) \rightarrow (U_{\infty,z_0}, z_0)$ in Caratheodory. After some argument, you only have to deal with $U_{\infty,z_0} \subset U_f$.

Now $U_{\infty,z_0}$ really depends on $z_0$ and sequence $f_n$. Different basepoint can lead to different Caratheodory limit. That is, given $z_0 \neq z_1$, we might have $U_{\infty, z_0} \neq U_{\infty, z_1}$. Similarly different sequence will give different limit. In my context, lets fix a sequence, so we only worry about base point.

Question: I want to say for all $z_0$, there exist $k>0$ so $f^k(U_{\infty,z_0}) = U_{\infty,z_0}$ i.e. eventually forward invariant domain. See picture below due to Arnaud Cheritat.

Here $(U_\infty,z_0)$ is the bottom left region, gets mapped to top right, which stays forward invariant

Now, forgetting Caratheodory topology and only use Hausdorff topology, we can say $U_{f_n} \rightarrow V$ in Hausdorff where $V$ open. We also can argue $V \subsetneq U_f$. Because $f_n(U_{f_n}) = U_{f_n}$, we immediately get $f(V)=V$, giving a sub-dynamical system of parabolic basin.

Perhaps easier to transfer the picture to $\mathbb{D}$ disk, so uniformize it and see we have a sub-dynamical system of a doubly parabolic Blaschke product. What are the possibility for such sub-dynamics?