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Box dimension of trajectories of some discrete dynamical systems

We study the asymptotics, box dimension, and Minkowski content of trajectories of some discrete dynamical systems. We show that under very general conditions, trajectories corresponding to parameters where saddle-node bifurcation appears have box dimension equal to~$1/2$, while those corresponding to period-doubling bifurcation parameter have box dimension equal to~$2/3$. Furthermore, all these trajectories are Minkowski nondegenerate. The results are illustrated in the case of logistic map.

logistic map; discrete dynamical system; box dimension; Minkowski content; bifurcation

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