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Thread: Dynamics on a circle

  1. #1
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    Dynamics on a circle

    Consider a circle $\displaystyle S$ and a function $\displaystyle f: S \mapsto S$ given by $\displaystyle f(\theta)=2 \theta \ (mod\ 2 \pi)$ where $\displaystyle \theta \in [0,2 \pi)$.

    Question: Given any open sets $\displaystyle U \in S$ and $\displaystyle V \in S$, how can we show that $\displaystyle \exists\ k$ such that $\displaystyle {f}^{k}(U)\cap V\neq \emptyset$?

    The problem comes from An Introduction to Chaotic Dynamical Systems by Devaney, Example 8.6. There's no explanation there though.
    Last edited by godelproof; Jan 1st 2012 at 02:36 AM.
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    Re: Dynamics on a circle

    Consider another problem:

    $\displaystyle {a}_{i}(x):=ix \ (mod\ 1)$, for $\displaystyle i=1,2,3,...$

    Now choose any irrational $\displaystyle x \in [0,1]$, prove that the series $\displaystyle \{{a}_{i}(x)\}$ is dense in $\displaystyle [0,1]$.

    There's a nice proof by mapping $\displaystyle {a}_{i}(x)$ to a circle of unit length, instead of considering mod 1 (In the same book by Devaney). But is there a more direct or simpler method? Because this seems like some classical problem.

    -------------------------------------------------------------------------------------
    Edit: If, say, $\displaystyle {a}_{i}(x):={2}^{i}x \ (mod\ 1)$, then the series no longer needs to be dense in $\displaystyle [0,1]$ for irrational x. So linearality is the key.
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    Re: Dynamics on a circle

    Quote Originally Posted by godelproof View Post
    Question: Given any open sets $\displaystyle U \in S$ and $\displaystyle V \in S$, how can we show that $\displaystyle \exists\ k$ such that $\displaystyle {f}^{k}(U)\cap V\neq \emptyset$?
    Do you mean $\displaystyle U\subseteq S$ and $\displaystyle V\subseteq S$?It is necessary that U and V are nonempty. Then U contains an interval. Repeatedly applying f to this interval inflates it so that eventually it covers the whole circle.
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    Re: Dynamics on a circle

    Quote Originally Posted by godelproof View Post
    There's a nice proof by mapping $\displaystyle {a}_{i}(x)$ to a circle of unit length, instead of considering mod 1 (In the same book by Devaney). But is there a more direct or simpler method? Because this seems like some classical problem.
    I would not consider dealing with a unit circle to be a different method. Considering a circle vs numbers mod 1 is just a reformulation of the problem.

    This is indeed a classic problem. A stronger statement that $\displaystyle \{a_i(x)\}$ is not only dense but uniform is the equidistribution theorem. The original fact follows from the observation that $\displaystyle \inf\{m\cdot1+n\cdot x\mid m\cdot1+n\cdot x>0\mbox{ and }m,n\in\mathbb{Z}\}=0$ since this set does not have a minimum; otherwise, the minimum would be a common divisor of 1 and x and they would be commensurate. (Of course, the absence of a minimum by itself does not imply that the $\displaystyle \inf$ is 0; another simple step is needed.) See also this thread and this post. See also this statement on cut-the-knot (set a = b = 0). The fact also follows from the Dirichlet's approximation theorem.
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    Re: Dynamics on a circle

    Quote Originally Posted by emakarov View Post
    Do you mean $\displaystyle U\subseteq S$ and $\displaystyle V\subseteq S$?It is necessary that U and V are nonempty. Then U contains an interval. Repeatedly applying f to this interval inflates it so that eventually it covers the whole circle.
    But why must repeated application of $\displaystyle f$ inflates ANY interval to cover the whole circle? This does not seem obvious to me. Say the interval is $\displaystyle ({x}_{1},{x}_{2})$ where $\displaystyle {x}_{1},{x}_{2} \in (2 \pi/{2}^{n+1},2 \pi/({2}^{n+1}-1))$, then the interval does not expand to cover $\displaystyle S$ in $\displaystyle n$ applications of $\displaystyle f$ or less. But at the $\displaystyle (n+1)$th application, we may again have $\displaystyle {f}^{n+1}({x}_{1}),{f}^{n+1}({x}_{2}) \in (2 \pi/{2}^{n+1},2 \pi/({2}^{n+1}-1))$, which we started out with. Maybe I'm missing something obvious, but I need more explanation here, please.
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    Re: Dynamics on a circle

    Quote Originally Posted by godelproof View Post
    But why must repeated application of $\displaystyle f$ inflates ANY interval to cover the whole circle?
    This is because $\displaystyle f$ transforms the whole interval and not only the endpoints.

    Consider the following picture.



    Even though $\displaystyle f^n(x_1)=2^nx_1\mbox{ mod }2\pi$ and $\displaystyle f^n(x_2)=2^nx_2\mbox{ mod }2\pi$ are quite close, the interval $\displaystyle (2^nx_1, 2^nx_2)$ is mapped to $\displaystyle [0,2\pi]$ by the modulo function. The interval $\displaystyle (f^n(x_1),f_n(x_2))$ covers the whole circle as soon as $\displaystyle 2^nx_2-2^nx_1>2\pi$.
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    Re: Dynamics on a circle

    Thank you. BTW, what software did you use to plot the picture? It's neat.
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    Re: Dynamics on a circle

    Quote Originally Posted by emakarov View Post
    I would not consider dealing with a unit circle to be a different method. Considering a circle vs numbers mod 1 is just a reformulation of the problem.

    This is indeed a classic problem. A stronger statement that $\displaystyle \{a_i(x)\}$ is not only dense but uniform is the equidistribution theorem. The original fact follows from the observation that $\displaystyle \inf\{m\cdot1+n\cdot x\mid m\cdot1+n\cdot x>0\mbox{ and }m,n\in\mathbb{Z}\}=0$ since this set does not have a minimum; otherwise, the minimum would be a common divisor of 1 and x and they would be commensurate. (Of course, the absence of a minimum by itself does not imply that the $\displaystyle \inf$ is 0; another simple step is needed.) See also this thread and this post. See also this statement on cut-the-knot (set a = b = 0). The fact also follows from the Dirichlet's approximation theorem.

    Can we find a real function $\displaystyle T: \mathbb{R} \mapsto \mathbb{R}$, such that the sequence {$\displaystyle T(\theta)\ mod \ 1, {T}^{2}(\theta)\ mod \ 1, {T}^{3}(\theta)\ mod \ 1,...$} is dense in $\displaystyle [0,1]$ whenever $\displaystyle \theta \in [0,1]$ is irrational?
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    Re: Dynamics on a circle

    Quote Originally Posted by godelproof View Post
    BTW, what software did you use to plot the picture? It's neat.
    I used PGF/TikZ (on CTAN and on sourceforge.net).

    Quote Originally Posted by godelproof View Post
    Can we find a real function $\displaystyle T: \mathbb{R} \mapsto \mathbb{R}$, such that the sequence {$\displaystyle T(\theta)\ mod \ 1, {T}^{2}(\theta)\ mod \ 1, {T}^{3}(\theta)\ mod \ 1,...$} is dense in $\displaystyle [0,1]$ whenever $\displaystyle \theta \in [0,1]$ is irrational?
    This seems plausible, but I am not a specialist in this area. Maybe you can make a new post (probably in the Calculus/Topology section) to attract attention to this question.
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