2021/06/22 by Neklyudov, Mikhail
#11B83 #37A05 #37A44 #Dynamical Systems (math.DS) #FOS: Mathematics #Functional Analysis (math.FA) #Number Theory (math.NT)
paper · doi:10.48550/arxiv.2106.11859
We investigate the problems related to the Collatz map T from the point of view of functional analysis. We associate with T certain linear operator T and show that cycles and (hypothetical) diverging trajectory (generated by T) correspond to certain classes of fixed points of operator T. Furthermore, we demonstrate connection between dynamical properties of operator T and map T. We prove that absence of nontrivial cycles of T leads to hypercyclicity of operator T. In the second part we show that the index of operator Id-T\inL(H2(D)) gives upper estimate on the number of cycles of T. For the proof we consider the adjoint operator F=T^* F: g→ g(z2)+\fracz-(1)/(3)3(g(z(2)/(3))+e(2πi)/(3)g(z(2)/(3)e(2πi)/(3))+e(4πi)/(3)g(z(2)/(3)e(4πi)/(3))), first introduced by Berg, Meinardus in \citeBM1994, and show it does not have non-trivial fixed points in H2(D). Moreover, we calculate resolvent of operator F and as an application deduce equation for the characteristic function of total stopping time σ∞. Furthermore, we construct an invariant measure for T in a slightly different setup, and investigate how the operator T acts on generalized arithmetic progressions.