A study on occurrence rates per bit for the logistic map over integers

We are investigating the logistic map over integers with finite precisions and sequences out-putted by the iterative mapping. In our proposed simple pseudo-random number generator with the logistic map over integers, we implicitly assumed that 0 and 1 occurrence frequencies per bit of outputs by the map are nearly equal. By some numerical experiments, we got the conjecture that occurrence rates per bit are nearly equal for many bits of 2n-bit values in the intermediate step of the map by one iteration and many iterations.