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context/motivation



if the angle on a calculator is set to radians, then it is very easy to demonstrate that iteration of $cos x$ (for arbitrary initial x) converges - simply keep pressing the $cos$ button! this unique fixed point $\alpha$ might reasonably be expected to be a transcendental number. (perhaps the answer to that is already known?). the conjecture outlined here suggests that $\alpha$ is an upper bound for a whole family of numbers defined in terms of iteration of particular sequences of cosine and sine functions. since these mixed iterations give rise to limit cycles rather than fixed points, we use the Cesaro mean to give a characteristic number for each cycle. the cycles i initially considered are easily defined in terms of the periodic binary representation of fractions whose denominator is not a power of 2. however whilst these are the only numbers corresponding to the sequences of sine and cosine that converge towards stable orbits, it seems likely that the periodicity itself is not the key factor ensuring Cesaro convergence, but that this is achieved due to a weaker asymptotic density condition which is necessary but not sufficient for periodicity. i apologise for any mistakes or lack of clarity in my (necessarily brief) presentation. the basic idea is simpler than may appear from a first glimpse of the definitions.

preliminary definitions


let $I$ be the closed unit interval $[0,1]$ so that the sine and cosine functions restrict to injective maps of $I$ into itself.
for integers $n \gt 0$ define $\beta_n:I \rightarrow \{0,1\}$ to be the $n^{th}$ binary digit of its argument, so $\beta_n(\lambda)=\lfloor 2^n\lambda \rfloor$
now define $\psi:\{0,1\} \times I \rightarrow I $ by: $$ \psi(0,\theta) = cos \theta \\ \psi(1,\theta) = sin \theta $$

every $\lambda \in I$ can be associated with a function $\Psi_{\lambda}:I \rightarrow I^{\omega}$ which generates a sequence in $I$, i.e.

$$ \forall \theta \in I, \Psi_{\lambda}(\theta) = \{\theta_n\}_{n=0,1,2,...} $$

with $\theta_0=\theta$ and for $n \ge 0$ $$ \theta_{n+1} = \psi(\beta_{n+1}(\lambda),\theta_n) $$

let us now call $\lambda \in I$ a $\beta$-number if an asymptotic density of $1$s in its binary representation exists, i.e. if the sequence $\{\beta_n(\lambda)\}$ has a Cesaro-mean. this mean, if it exists, we may denote by $\beta^*(\lambda)$

let us also define $\alpha$ as the unique fixed point in I of the cosine function, i.e.

$$cos \; \alpha = \alpha$$

conjecture


1. $\forall \theta \in I$ the sequence $\Psi_{\lambda}(\theta)$ has a Cesaro mean if and only if $\lambda$ is a $\beta$-number, and in this case the Cesaro mean is independent of $\theta$ and may be denoted $\Psi_{\lambda}^*$

2. if the sequence $\Psi_{\lambda}(\theta)$ has Cesaro mean, then this mean is equal to $\alpha$ if and only if $\beta^*(\lambda)=0$

3. for any $\beta$-number $\lambda$ if $\beta^*(\lambda) \gt 0$ then $\Psi_{\lambda}^*\lt \alpha$

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