We consider thus the functional notion of communication within the theory of dynamic systems, where the finite difference equation, or flow of a differential equation, is iterated upon as a function. Within this framework, there is a true underlying deterministic system which can be explained by a single function that is iterated upon over time as the repeated trials of the experiment. The space of outcomes may be partitioned, often into intervals, and then coded into a symbolic system of numerical sequences, where the place-order indicates the number of iterations of the function and the place-value the indexed interval it falls within [1]. This classifies initial conditions by their results, and may even represent a homomorphism between initial points and resulting codes.

The proper functional notion of a probabilistic system is a \textbf{dysfunction} since it sends one state to many possible different states, in the sense that obtains an uncountably many values at 0 since analytically , i.e. the whole unit interval. We may study these as rather iterated functional recursions, where when there is a sensitive dependence upon initial conditions, we will likely find the function to be \textbf{hyperbolic} in that the derivatives of these iterates grow at least geometrically, and the problem of dysfunctionality will become one of imprecision in initial conditions.

From the previous lecture, a Markov chain is a dynamic system where, for n independent variables [2]. While the system may be \say{fully determined} by an initial condition , i.e. the exact (pre-partitioned) system at time t is given by , limits on the precision of measurement prevent us from ever either determining empirically or choosing an exact value to begin iterations upon . Thus, the secondary independent random variable

Given a Partitioning

Consider the modeling of a communication system. A message is sent through this system, arriving at a state of the system at each time. The message is thus the temporal reconfiguration chain of the system as it takes on different states from its possibility set

From the stochastic process approach, we might ask thus about the stationary distribution of these sampling codes, as what are the initial distribution conditions such that the distribution positions do not change over time? Let

Consider the string of numbers

**References**

- Robert L. Devaney,
*An Introduction to Chaotic Dynamical Systems*, 2nd Ed. Addison-Wesley Publishing Company, Inc, The Advanced Book Program, 1948. - Richard Serfozo,
*The Basics of Applied Stochastic Processes*, Probability and its Applications. Springer-Verlag Berlin Heidelberg, 2009. p. 1.

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