Consider a Cobweb diagram on the following function in the interval $(\frac12,1]$:
The function is clearly arranged as a fractal, in fact its most obvious symmetry is $x\cong \dfrac{x+2}4$
By this symmetry alone, the function has infinitely many points which map to any given point. And in fact, this is the only symmetry by which the function does not surject. On other words, in the quotient by this symmetry, it surjects onto any given choice of representatives.
It seems to me to follow that the right hand part of the diagram $(\frac34,1]$ should be all we need in order to create the cobweb diagram.
You may notice that this entire right hand segment of the function is below the $y=x$ line (which I have not drawn on). (apologies, the $x$ axis is inadvertently scaled to be over-square, with the effect that the $y=x$ line wouldn't run at 45 degrees.)
Anyway. Am I right in thinking that this latter fact, i.e. the function lying entirely below the 45 degree line in this representative interval, is sufficient to guarantee the function has no periodic points?
If so, this is sufficient to prove Lagarias' periodicity conjecture, since this graph is topologically conjugate to the Collatz conjecture function.
