Dynamics of exponential linear map in functional space

We consider the question of existence of a unique invariant probability distribution which satisfies some evolutionary property. The problem arises from the random graph theory but to answer it we treat it as a dynamical system in the functional space, where we look for a global attractor. We consider the following bifurcation problem: Given a probability measure $μ$, which corresponds to the weight distribution of a link of a random graph we form a positive linear operator $Φ$ (convolution) on distribution functions and then we analyze a family of its exponents with a parameter $λ$ which corresponds to connectivity of a sparse random graph. We prove that for every measure $μ$ (\emph{i.e.}, convolution $Φ$) and every $λ< e$ there exists a unique globally attracting fixed point of the operator, which yields the existence and uniqueness of the limit probability distribution on the random graph. This estimate was established earlier \cite{KarpSipser} for deterministic weight distributions (Dirac measures $μ$) and is known as $e$-cutoff phenomena, as for such distributions and $λ>e$ there is no fixed point attractor. We thus establish this phenomenon in a much more general sense.

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