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Math Help - Dynamics on a circle

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    Dynamics on a circle

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

    Question: Given any open sets U \in S and V \in S, how can we show that \exists\ k such that {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; January 1st 2012 at 02:36 AM.
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    Re: Dynamics on a circle

    Consider another problem:

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

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

    There's a nice proof by mapping {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, {a}_{i}(x):={2}^{i}x \ (mod\ 1), then the series no longer needs to be dense in [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 U \in S and V \in S, how can we show that \exists\ k such that {f}^{k}(U)\cap V\neq \emptyset?
    Do you mean U\subseteq S and 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 {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 \{a_i(x)\} is not only dense but uniform is the equidistribution theorem. The original fact follows from the observation that \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 \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 U\subseteq S and 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 f inflates ANY interval to cover the whole circle? This does not seem obvious to me. Say the interval is ({x}_{1},{x}_{2}) where {x}_{1},{x}_{2} \in (2 \pi/{2}^{n+1},2 \pi/({2}^{n+1}-1)), then the interval does not expand to cover S in n applications of f or less. But at the (n+1)th application, we may again have {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 f inflates ANY interval to cover the whole circle?
    This is because f transforms the whole interval and not only the endpoints.

    Consider the following picture.



    Even though f^n(x_1)=2^nx_1\mbox{ mod }2\pi and f^n(x_2)=2^nx_2\mbox{ mod }2\pi are quite close, the interval (2^nx_1, 2^nx_2) is mapped to [0,2\pi] by the modulo function. The interval (f^n(x_1),f_n(x_2)) covers the whole circle as soon as 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 \{a_i(x)\} is not only dense but uniform is the equidistribution theorem. The original fact follows from the observation that \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 \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 T: \mathbb{R} \mapsto \mathbb{R}, such that the sequence { T(\theta)\ mod \ 1, {T}^{2}(\theta)\ mod \ 1, {T}^{3}(\theta)\ mod \ 1,...} is dense in [0,1] whenever \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 T: \mathbb{R} \mapsto \mathbb{R}, such that the sequence { T(\theta)\ mod \ 1, {T}^{2}(\theta)\ mod \ 1, {T}^{3}(\theta)\ mod \ 1,...} is dense in [0,1] whenever \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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