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The dynamics of one dimensional iterative maps in the regime of fullydeveloped chaos is studied in detail. Motivated by the observation of dynamicalstructures around the unstable fixed point we introduce the geometrical conceptof a turning point which represents a local minimum or maximum of thetrajectory. Following we investigate the highly organized and structureddistribution of turning points. The turning point dynamics is discussed and thecorresponding turning point map which possesses an appealing asymptotic scalingproperty is investigated. Strong correlations are shown to exist for theturning point trajectories which contain the information of the fixed points aswell as the stability coefficients of the dynamical system. For the morespecialized case of symmetric maps which possess a symmetric density we deriveuniversal statistical properties of the corresponding turning point dynamics.Using the turning point concept we finally develop a method for the analysis of(one dimensional) time series.Comment: 37 pages, 11 figures, LaTeX, tccite.sty, to be published in the Int.  J. Bif. Chao

A Turning Point Analysis of the Ergodic Dynamics of Iterative Maps