is, the periodic orbits of the map are in one-to-one correspondence with the rationals. This phenomenon is note-worthy, because something similar happens in many chaotic systems. For example, geodesics on compact manifolds can have periodic orbits that behave in this way. Keep in mind, however, that the rationals are a set of measure zero in the reals. Almost all orbits are not periodic! The aperiodic orbits correspond to the irrational numbers. This property also holds true in a more general setting. An open question is to what degree the behavior of the periodic orbits constrain the behavior of the system as a whole. Phenomena such as Arnold diffusion suggest that the general answer is "not very much". Density formulation Instead of looking at the orbits of individual points under the action of the map, it is equally worthwhile to explore how the map affects densities on the unit interval. That is, imagine sprinkling some dust on the unit interval; it is denser in some places than in others. What happens to this density as one iterates? Write as this density, so that . To obtain the action of on this density, one needs to find all points and write The denominator in the above is the Jacobian determinant of the transformation, here it is just the derivative of and so . Also, there are obviously only two points in the preimage of , these are and Putting it all together, one gets By convention, such maps are denoted by so that in this case, write The map is a linear operator, as one easily sees that and for all functions on the unit interval, and all constants . Viewed as a linear operator, the most obvious and pressing question is: what is its spectrum? One eigenvalue is obvious: if for all then one obviously has so the uniform density is invariant under the transformation. This is in fact the largest eigenvalue of the operator , it is the Frobenius–Perron eigenvalue. The uniform density is, in fact, nothing other than the invariant measure of the dyadic transformation. To explore the spectrum of in greater detail, one must first limit oneself to a suitable space of functions (on the unit interval) to work with. This might be the space of Lebesgue measurable functions, or perhaps the space of square integrable functions, or perhaps even just polynomials. Working with any of these spaces