Discrete dynamics of one dimensional Collatz like integral value transformations
In Discrete dynamics and number theory, one of the most important challenging conjectures is Collatz conjecture. Some basic algebraic and analytical properties of Collatz like integral value transformations (IVTs) in the context of discrete dynamical system over \(\mathbb {N}_0\) are adumbrated. The dynamical maps associated to the dynamical systems are everywhere continuous but nowhere differentiable (smooth) in domain \(\mathbb {N}_0\). Under such Collatz like IVTs, for any initial point \(X_0\) \(\in \) \(\mathbb {N}_0\) the dynamical systems are convergent and converge eventually to a single point attractor, zero. In other words, the image sets associated to the Collatz like dynamical maps in each iteration converge to a single point image set consisting the point zero. The rate of convergence of the Collatz like dynamical system with respect to some property is also described. It is observed that the \(l_2\)-norm of the ith order derivative operator for all Collatz like map is a decreasing convergent sequence which converges to zero.
discrete dynamical systems fractal dimension integral value transformations & collatz like ivts 37d45 37m99 34a30