Baker transformation as autoregression system

We study the baker transformation in the context of an autoregression model (digital filter) of the first order. An initial condition x₀ is supposed to be a random value having the uniform distribution on the interval (0,1). Being unbiased, binary digits of x₀, 0 and 1, have the occurrence probability equal to 1/2 . The y-component of the baker transformation is represented as a linear autoregression equation of the first order where binary digits of x₀ play the role of an excitation (input signal). It is shown that the digital filter corresponding to the baker transformation is causal, stable and reversible one. The asymptotic regime of baker transform dynamics does not depend on the distribution of the initial value y₀.