Enveloping algebras and dynamical systems from automata: With comparisons to statistical methods and with applications to gene expression

Automata from topological subshifts are investigated.The index set of the automata is newly defined from the directed acyclic graphs, whose trees are studied after substochastic probability matrixes, from marginalization of the opportune variables after Tucker-like method, of the new suitable accumulation, time-dependent model. The automata are described after a dynamical system to which the Tomita-Takesaki theory is proven to be applied.The Rokhlin aperiodic endomorphism is applied the Margulis foliation theory on the open topological set;The techniques of the new foliation allow one to relate the subalgebras of the *-algebra on the Hilbert space with the Radon measure on the Banach l1 space:the new Miatello-Wallach meromorphic function is defined, after which the variances are calculated after potential theory. The dynamical system is a one-dimensional one on a torus, with characteristic Ljapunov exponenet equals 0, as here newly proven after upgradingthe Oseledets multiplicative ergodic theorem of dynamical systems, for which the definition of forward stable partition holds.Statistical methods for graph theory are proposed, and methodologies in gene expression are individuated.