Wada Lakes are three disjoint open subsets of $\mathbb R^2$ with common boundary. Originally they were constructed by hand, but they also arise naturally in the real life, that is, theory of dynamical systems. (See, for example, this <http://www.math.cornell.edu/~hubbard/pendulum.pdf> paper of John Hubbard.)


From dynamical construction it is clear that one can have a homeomorphism that permutes the lakes. For example, this expository paper <http://www.ams.org/notices/200601/fea-coudene.pdf> gives such a homeomorphism (in fact, a diffeomorphism) for lakes on the sphere $S^2$. The dynamics on the boundary of the lakes is chaotic. 


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Question 1: Let $U\_1$, $U\_2$ and $U\_3$ be open subset of $\mathbb R^d$ with common boundary $C$. Assume that each $U\_i$ is an image of an injective continuous map $\mathbb R^d\to\mathbb R^d$. Is there a homeomorphism of $\mathbb R^d$ that permutes open sets $U\_i$ and is identity on $C$?


Question 2: Let $U\_i\subset \mathbb R^d$, $i\in \mathbb Z$, be a disjoint collection of open sets each of which is homeomorphic to $\mathbb R^d$. Is there a homeomorphism of $\mathbb R^d$ that satisfies the following


* $h$ maps $U\_i$ onto $U\_{i+1}$
* every point in $\partial U\_i$ is periodic under $h$


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**Edit:** Below André Henriques has produced two examples when such a homeomorphism does not exist. Both examples make use of some local structure that cannot be permuted. It is still not clear what happens for "abstact Wada Lakes": are these examples exceptions or arbitrary Wada Lakes cannot be permuted as well? 


**Edit2** (Jan. 9th 2012) 


* In is clear now that there is no such homeomorphism if the dimension $d=2$. (See the comments below).
* The answer to Q1 is also negative in higher dimension as explained in the reference given by Andres Koropecki.
* For $d\ge 3$ there are examples by André Henriques of open sets for which one cannot find such a homeomorphism. In general, Question 2 is still open.



Permute Wada Lakes keeping the coastline intact? (still open in dim >2)
In dimension 2, the answer is also "no".


Recall the classical construction of the [Wada lakes](http://en.wikipedia.org/wiki/File:Lakes_of_Wada.jpg).
In the 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.




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


In that case, there is no homeomorphism of $\mathbb{R}^2$ 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 $\cap V$ and Green-Lake $\cap V$ intersect $\partial V$ in exactly two intervals (taking $V$ be a metric ball will do).
* For every sufficiently small neighborhood $V$ of $P$, there exist connected components of
Blue-Lake $\cap V$ that intersect $\partial V$ in at least three intervals (the components containing the straights).


