Permute Wada Lakes keeping the coastline intact? (still open in dim >2) - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-23T18:45:34Z http://mathoverflow.net/feeds/question/34651 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/34651/permute-wada-lakes-keeping-the-coastline-intact-still-open-in-dim-2 Permute Wada Lakes keeping the coastline intact? (still open in dim >2) Andrey Gogolev 2010-08-05T16:27:21Z 2012-02-17T20:45:49Z <p>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 <a href="http://www.math.cornell.edu/~hubbard/pendulum.pdf" rel="nofollow">http://www.math.cornell.edu/~hubbard/pendulum.pdf</a> paper of John Hubbard.)</p> <p>From dynamical construction it is clear that one can have a homeomorphism that permutes the lakes. For example, this expository paper <a href="http://www.ams.org/notices/200601/fea-coudene.pdf" rel="nofollow">http://www.ams.org/notices/200601/fea-coudene.pdf</a> 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. </p> <p>==</p> <p>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 $C$?</p> <p>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</p> <ul> <li>$h$ maps $U_i$ onto $U_{i+1}$</li> <li>every point in $\partial U_i$ is periodic under $h$</li> </ul> <p>==</p> <p><strong>Edit:</strong> 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? </p> <p><strong>Edit2</strong> (Jan. 9th 2012) </p> <ul> <li>In is clear now that there is no such homeomorphism if the dimension $d=2$. (See the comments below).</li> <li>The answer to Q1 is also negative in higher dimension as explained in the reference given by Andres Koropecki.</li> <li>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. </li> </ul> http://mathoverflow.net/questions/34651/permute-wada-lakes-keeping-the-coastline-intact-still-open-in-dim-2/34940#34940 Answer by André Henriques for Permute Wada Lakes keeping the coastline intact? (still open in dim >2) André Henriques 2010-08-08T19:28:39Z 2012-02-17T20:42:42Z <p>The answer to your question is "no".</p> <p>I'm working in $\mathbb{R}^3$. Let $C_1, C_2$ be open balls whose boundaries touch along an interval $J := \partial C_1 \cap \partial C_2$, and let $P \in J$ be a point on that interval. For concreteness:</p> <p>$C_1 := ]0,1[ \times ]0,1[ \times ]0,1[ = (0,1)^3$</p> <p>$C_2:= ]1,2[ \times ]1,2[ \times ]0,1[$</p> <p>$J := \lbrace 1\rbrace \times \lbrace 1\rbrace \times ]0,1[$,</p> <p>and $P := (1, 1, 0.5).$</p> <p>It is possible to construct $U_1, U_2, U_3 \subset \mathbb{R}^3$ with common boundary, such that</p> <ul> <li>$C_1\subset U_1$</li> <li>$C_2 \subset U_2$</li> <li>No $2$-disk in $U_3$ bounds the interval $J$.</li> </ul> <p>The above properties exclude the possibility of a homeomorphism that fixes $P$ and exchanges $U_3$ with another $U_i$. <hr> I now describe the open sets $U_1, U_2, U_3$ in a neighborhood of $P$.</p> <p>Let $Q := ]1,2[ \times ]0,1[ \times ]0,1[$ and $R := ]0,1[ \times ]1,2[ \times ]0,1[$.</p> <p>I'll describe what $U_1, U_2, U_3$ look like inside $Q$ and $R$, and leave the rest of the construction unspecified. Actually, the construction will be identical in $Q$ and in $R$, so I'll just describe the construction in $Q$.</p> <p>Let $J' \subset J$ be a countable dense subset, $J' = \lbrace j_1, j_2, j_3, j_4, \dots \rbrace$, and let $Q' \subset Q$ be a countable dense subset $Q' = \lbrace q_1, q_2, q_3, q_4, \dots \rbrace$. Let $F:= \partial Q$ \ $(\partial C_1 \cup \partial C_2)$ be the part of the boundary of $Q$ that doesn't touch the other cubes. We inductively pick "cones" $V_n \subset Q$ with the following properties:</p> <ul> <li>$V_n$ is homeomorphic to a 3-ball</li> <li>$j_n \in \partial V_n$, and it is the unique point of $J$ in the boundary of $V_n$</li> <li>$V_n \cap F$ is homeomorphic to a $2$-disc.</li> <li>The closure of $V_n$ doesn't intersect the closures of the other $V_m$'s.</li> <li>Unless $q_n$ is already in the closure of some other $V_m, m &lt; n$, we make sure that $q_n \in V_n$. </li> </ul> <p>One can then partition the $V_n$'s into three classes: $V_{3k+1}, V_{3k+2}, V_{3k+3}$. The open sets $U_i$ will be such that </p> <p>$$U _ i \cap Q = \bigcup_{k\ge 1} V _ {3k+i}.$$</p> <p>Unless one does something very stupid, the subsets $U_i \cap Q$ will all have the same boundary within $Q$.</p> http://mathoverflow.net/questions/34651/permute-wada-lakes-keeping-the-coastline-intact-still-open-in-dim-2/34993#34993 Answer by André Henriques for Permute Wada Lakes keeping the coastline intact? (still open in dim >2) André Henriques 2010-08-09T11:40:48Z 2012-02-17T20:45:49Z <p>In dimension 2, the answer is also "no".</p> <p>Recall the classical construction of the <a href="http://en.wikipedia.org/wiki/File:Lakes_of_Wada.jpg" rel="nofollow">Wada lakes</a>. 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.</p> <hr> <p>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:</p> <ul> <li>The blue straights forms a converging sequence with limit point $P$.</li> <li>The red and green straights all lie outside $U$.</li> </ul> <p>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:</p> <ul> <li>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).</li> <li>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).</li> </ul> http://mathoverflow.net/questions/34651/permute-wada-lakes-keeping-the-coastline-intact-still-open-in-dim-2/81391#81391 Answer by Andres Koropecki for Permute Wada Lakes keeping the coastline intact? (still open in dim >2) Andres Koropecki 2011-11-20T02:16:36Z 2012-02-17T20:43:40Z <p>I'm late by a year, but just in case, the result from this one-page paper seems to answer your question (but it has nothing to do with the Wada property):</p> <p>M. Brown and J. M. Kister, Invariance of complementary domains of a fixed point set, Proc. Amer. Math. Soc. 91 (1984), no. 3, 503–504. MR 744656</p> <p>The theorem is as follows: Let $f$ be a homeomorphism of a connected topological manifold $M$ with fixed point set $F$. Then either (1) each connected component of $M-F$ is invariant, or (2) there are exactly two components and $f$ interchanges them.</p>