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In dimension 2, the answer is also "no".

Recall the classical construction of the Wada lakes. In the linked picture, one sees little "straits" connecting the red/blue/green regions at stage n$n$ with the extra windy strip that is added at stage n+3. $n+3$. For example, one sees a blue strait roughly in the middle of the picture, and a red strait (barely visible) on the upper left part. The location of these straights are free parameters in the construction of the Wada lakes.

Now I proceed to construct the desired counterexample. Let P $P$ be a point on the boundary of the yet-to-be-constructed Wada lakes. And let U $U$ be a fixed ball around P. $P$. The straights can be picked so that:

  • The blue straights forms a converging sequence with limit point P.$P$.
  • The red and green straights all lie outside U.$U$.

In that case, there is no homeomorphism of ℝ2$\mathbb{R}^2$ fixing P, $P$, and exchanging the blue lake with a lake of another color. The reason is the following:

  • There exist arbitrarily small neighborhoods V $V$ of P $P$ such that all connected components of Red-Lake ∩ V $\cap V$ and Green-Lake ∩ V $\cap V$ intersect ∂V $\partial V$ in exactly two intervals (taking V $V$ be a metric ball will do).
  • For every sufficiently small neighborhood V $V$ of P, $P$, there exist connected components of Blue-Lake ∩ V $\cap V$ that intersect ∂V $\partial V$ in at least three intervals (the components containing the straights).
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In dimension 2, the answer is also "no".

Recall the classical construction of the Wada lakes: . In the above linked picture, one sees little "straits" connecting the red/blue/green regions at stage n with the extra windy strip that is added at stage n+3. For example, one sees a blue strait roughly in the middle of the picture, and a red strait (barely visible) on the upper left part. The location of these straights are free parameters in the construction of the Wada lakes.

Now I proceed to construct the desired counterexample. Let P be a point on the boundary of the yet-to-be-constructed Wada lakes. And let U be a fixed ball around P. The straights can be picked so that:

  • The sequence of blue straights forms a converging sequence with limit point P.
  • The red and green straights all lie outside U.

Then

In that case, there is no homeomorphism of ℝ2 fixing P, and exchanging the blue lake with a lake of another color. The reason is the following:

  • There exist arbitrarily small neighborhoods V of P such that all connected components of Red-Lake ∩ V and Green-Lake ∩ V intersect ∂ V ∂V in exactly two intervals only(taking V be a metric ball will do).
  • For every sufficiently small neighborhood V of P, there exists exist connected components of Blue-Lake ∩ V that intercect ∂ V intersect ∂V in at least four three intervals (the components containing the straights).
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In dimension 2, the answer is also "no".

Recall the classical construction of the Wada lakes: Wada Lakes

In the above picture, one sees little "straits" connecting the red/blue/green regions at stage n with the extra windy strip that is added at stage n+3. For example, one sees a blue strait roughly in the middle of the picture, and a red strait (barely visible) on the upper left part. The location of these straights are free parameters in the construction of the Wada lakes.

Now I proceed to construct the desired counterexample. Let P be a point on the boundary of the yet-to-be-constructed Wada lakes. And let U be a fixed ball around P.

The straights can be picked so that:

  • The sequence of blue straights forms a converging sequence with limit point P.
  • The red and green straights all lie outside U.

Then, there is no homeomorphism of ℝ2 fixing P, and exchanging the blue lake with a lake of another color. The reason is the following:

  • There exist arbitrarily small neighborhoods V of P such that all connected components of Red-Lake ∩ V and Green-Lake ∩ V intersect ∂ V in two intervals only.
  • For every sufficiently small neighborhood V of P, there exists connected components of Blue-Lake ∩ V that intercect ∂ V in at least four intervals.