Fractals play an important and very interesting role in science. Modeling dynamical processes on fractal sets as state space has been challenging because of the absence of smooth structures. In order to overcome that difficulty we work with a space which is "equivalent" to our fractal set in the topolological sense. Here comes the concept of homeomorphism. More precisely, suppose $ f: X→Y$ is a bijective (one-to-one and onto) function between topological spaces $ X$ and $ Y$. Since $ f$ is bijective, the inverse $ f^{-1}$ exists. If both $ f$ and $ f^{-1}$ are continuous, then $ f$ is called a homeomorphism. We then write $X≅Y.$ Here are few examples:

Having established the homeomorphism, the idea is to work with the space which is better known to us, for which we have a rich theory. This enables us to carry-over many topological, measure theoretical, and dynamical properties and behaviors from one space to the other. In my dissertation, the two spaces I considered are (1) a certain class of fractal sets, called Julia sets, and (2) the Martin boundary of a suitably defined Markov chain having the word-space (from a finite alphabet) as its state space.

What is Martin boundary? Why should we consider Martin boundary? The classical Poisson formula yields an integral representation of a bounded harmonic function in the unit disk in the complex plane $ℂ$ in terms of its boundary values: Suppose $u(z)$ is harmonic in a domain containing the disc $|z|≤ R.$ Then for $z=re^{iθ},$ $r\< R,$ in the unit disc we have (Poisson integral formula) $$u(re^{iθ})=1/{2π} ∫_{0}^{2π} {R^2-r^2}/{R^2-2rR\,cos(θ-φ)+r^2}\, u(Re^{iφ})\,dφ$$ This kind of result can be generalized for more complicated spaces such as a fractal set. Given a Markov operator $P$ on a state space $X$, we can easily define harmonic functions as invariant functions of the operator $P,$ but in order to speak about their boundary values we need a boundary. Since no boundary is normally attached to the state space of a Markov chain (as distinct from bounded Euclidean domains common for the classical potential theory). One way to overcome this limitation is to find a topological compactification of the state space that is naturally connected with the Markov operator $P.$

In the dissertation, I have established a homeomorphism between a certain class of Julia sets (totally disconnected Julia sets) and the Martin boundary of a suitably defined Markov chain by using symbolic dynamics with a finite set of alphabets. When the Julia set is also bounded, this connection allows to relate various thermodynamic concepts – such as entropy, measure of maximal entropy, Gibbs measure, and measure of equilibrium – to potential theoretic concepts like capacity and harmonic measures on the Julia set. I have proved a number of results in this context.