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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?

• I don't know the answer to the same question even when dropping the condition $f|_C=id$. Note that the situation is rather different from Wada Lakes in $S^2$, since the common boundary is no longer compact.
• Motivation comes from partially hyperbolic dynamics and is quite intricate and long.
• Are there other contexts in which Wada Lakes appear?
• Is there a comprehensive topological survey of the subject? Mostly, I would like to know about properties of the boundary $C$. It must be indecomposable continuum, but I don't know much beyond this.
• Actual question I am interested in is different, but close in spirit. I have countably many disjoint open sets in $\mathbb R^d$. These open sets are homeomorphic to open balls. There is a homeomorphism $h$ that permutes ($U_i\to U_{i+1}$) open sets and pointwise periodic on $\partial U_i$ for all $i$ (I don't think there's a result that allows me to conclude that $h$ is actually periodic on $\partial U_i$). Also, I know that $\partial U_i$ is a two-sided repeller for $h$. I would like to show that this crazy senario is, in fact, impossible.

Edit2 (Dec. 12th 2011Jan. 9th 2012)

• In is clear now that there is no such homeomorphism if the dimension $d=2$. (See the comments below and below).
• The answer to Q1 is also negative in higher dimension as explained in the reference from 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, it Question 2 is still open. That is, does there exist Wada Lakes that admit "permuting homeomorphism"?

• I have a nice application so I am still interested in the problem (the more complicated version stated in the last comment above).

• 5 added 531 characters in body; edited title

# Permute Wada Lakes keeping the coastline intact? (stillopenindim>2)

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: 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 $C$?

==============================================

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?

• I don't know the answer to the same question even when dropping the condition $f|_C=id$. Note that the situation is rather different from Wada Lakes in $S^2$, since the common boundary is no longer compact.
• Is there a comprehensive topological survey of the subject? Mostly, I would like to know about properties of the boundary $C$. It must be indecomposable continuum, but I don't know much beyond this.
• Actual question I am interested in is different, but close in spirit. I have countably many disjoint open sets with common boundary $C$ sitting in $\mathbb R^d$. These open sets are continuous images of an homeomorphic to open ball and unboundedballs. There is a homeomorphism $h$ that permutes ($U_i\to U_{i+1}$) open sets and pointwise periodic on $C$ \partial U_i$for all$i$(I don't think there's a result that allows me to conclude that$h$is actually periodic on$C$). \partial U_i$). Also, I know that $C$ \partial U_i$is a two-sided repeller for$h$. I would like to show that this crazy senario is, in fact, impossible. Edit2 (Dec. 12th 2011) • In is clear now that there is no such homeomorphism if the dimension$d=2$. (See the comments below and the reference from Andres Koropecki. • For$d\ge 3$there are examples by André Henriques of open sets for which one cannot find such a homeomorphism. In general, it is still open. That is, does there exist Wada Lakes that admit "permuting homeomorphism"? • I have a nice application so I am still interested in the problem (the more complicated version stated in the last comment above). 4 added 149 characters in body Wada Lakes are three disjoint open subsets of$\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$. The dynamics on the boundary of the lakes is chaotic. ============================================== Question: Let$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$? ============================================== Edit: Below André Henriques has produced an example two examples when such a homeomorphism does not exist. This works for$d\ge 3$Both examples make use of some local structure that cannot be permuted. What happens in general It is still not clear .what happens for "abstact Wada Lakes": are these examples exceptions or arbitrary Wada Lakes cannot be permuted as well? Comments: • I don't know the answer to the same question even when dropping the condition$f|_C=id$. Note that the situation is rather different from Wada Lakes in$S^2$, since the common boundary is no longer compact. • Motivation comes from partially hyperbolic dynamics and is quite intricate and long. • Are there other contexts in which Wada Lakes appear? • Is there a comprehensive topological survey of the subject? Mostly, I would like to know about properties of the boundary$C$. It must be indecomposable continuum, but I don't know much beyond this. • Actual question I am interested in is different, but close in spirit. I have countably many disjoint open sets with common boundary$C$sitting in$\mathbb R^d$. These open sets are continuous images of an open ball and unbounded. There is a homeomorphism$h$that permutes ($U_i\to U_{i+1}$) open sets and pointwise periodic on$C$(I don't think there's a result that allows me to conclude that$h$is actually periodic on$C$). Also, I know that$C$is a two-sided repeller for$h\$. I would like to show that this crazy senario is, in fact, impossible.