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edited Jan 10 2012 at 0:29 |
=============================================== 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 $h$=============================================== 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? 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 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).
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edited Dec 13 2011 at 3:12 |
Permute Wada Lakes keeping the coastline intact? (still open in dim >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.
==============================================
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?
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 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).
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edited Aug 9 2010 at 20:28 |
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.
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edited Aug 9 2010 at 0:37 |
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 when such a homeomorphism does not exist. This works for $d\ge 3$. What happens in general is still not clear.
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.
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edited Aug 8 2010 at 4:43 |
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 $R^d$ that permutes open sets $U_i$ and is identity on $C$?
==============================================
Comments:
- Motivation comes from partially hyperbolic dynamics and is quite intricate and long.
- Is it possible
- I don't know the answer to permute them without any requirements on the dynamics of same question even when dropping the boundary? 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.
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asked Aug 5 2010 at 16:27 |
Permute Wada Lakes keeping the coastline intact?
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 $R^d$ that permutes open sets $U_i$ and is identity on $C$?
==============================================
Comments:
- Motivation comes from partially hyperbolic dynamics and is quite intricate and long.
- Is it possible to permute them without any requirements on the dynamics of the boundary? Note, the situation is rather different from Wada Lakes in $S^2$, since the common boundary is no longer compact.
- 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.
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