occurs. If we start with information on the first s bits of the initial iterate, then after m simulated iterations (m < s) we only have s − m bits of information remaining. Thus we lose information at the exponential rate of one bit per iteration. After s iterations, our simulation has reached the fixed point zero, regardless of the true iterate values; thus we have suffered a complete loss of information. This illustrates sensitive dependence on initial conditions—the mapping from the truncated initial condition has deviated exponentially from the mapping from the true initial condition. And since our simulation has reached a fixed point, for almost all initial conditions it will not describe the dynamics in the qualitatively correct way as chaotic. Equivalent to the concept of information loss is the concept of information gain. In practice some real-world process may generate a sequence of values (xn) over time, but we may only be able to observe these values in truncated form. Suppose for example that x0 = 0.1001101, but we only observe the truncated value 0.1001. Our prediction for x1 is 0.001. If we wait until the real-world process has generated the true x1 value 0.001101, we will be able to observe the truncated value 0.0011, which is more accurate than our predicted value 0.001. So we have received an information gain of one bit. Relation to tent map and logistic map The dyadic transformation is topologically semi-conjugate to the unit-height tent map. Recall that the unit-height tent map is given by The conjugacy is explicitly given by so that That is, This is stable under iteration, as It is also conjugate to the chaotic r = 4 case of the logistic map. The r = 4 case of the logistic map is ; this is related to the bit shift map in variable x by There is also a semi-conjugacy between the dyadic transformation (here named angle doubling map) and the quadratic polynomial. Here, the map doubles angles measured in turns. That is, the map is given by Periodicity and non-periodicity Because of the simple nature of the dynamics when the iterates are viewed in binary notation, it is easy to categorize the dynamics based on the initial condition: If the initial condition is irrational (as almost all points in the unit interval are), then the dynamics are non-periodic—this follows directly from the definition of an irrational