We uncover the dynamics at the chaos threshold ${\mu }_{\infty }$ of the logistic map and find that it consists of trajectories made of intertwined power laws that reproduce the entire period-doubling cascade that occurs for $\mu <{\mu }_{\infty }.$ We corroborate this structure analytically via the Feigenbaum renormalization-group (RG) transformation and find that the sensitivity to initial conditions has precisely the form of a q exponential, of which we determine the q index and the q-generalized Lyapunov coefficient ${\lambda }_q.$ Our results are an unequivocal validation of the applicability of the nonextensive generalization of Boltzmann-Gibbs statistical mechanics to critical points of nonlinear maps.

We look at the properties of clusters of order parameter Φ at critical points in thermal systems and consider their significance to statistical–mechanical ground rules. These properties have been previously obtained through the saddle-point approximation in a coarse-grained partition function. We examine both static and dynamical aspects of a single large cluster and indicate that these properties fall outside the canonical Boltzmann–Gibbs (BG) scheme. Specifically, (1) the faster than exponential growth with cluster size of the space-integrated Φ suggests non-extensivity of the BG entropy but extensivity of a q-entropy expression. (2) The finding that the time evolution of Φ is described by the dynamics of an intermittent nonlinear map implies an atypical sensitivity to initial conditions compatible with q-statistics and displays an ‘aging’ scaling property. (3) Both the approach to criticality and the infinite-size cluster limit at criticality manifest through a crossover from canonical to q-statistics and we discuss the non-uniform convergence associated with these features.

Transitions to chaos in archetypal low-dimensional nonlinear maps offer real and precise model systems in which to assess proposed generalizations of statistical mechanics. The known association of chaotic dynamics with the structure of Boltzmann–Gibbs (BG) statistical mechanics has suggested the potential verification of these generalizations at the onset of chaos, when the only Lyapunov exponent vanishes and ergodic and mixing properties cease to hold. There are three well-known routes to chaos in these deterministic dissipative systems, period-doubling, quasi-periodicity and intermittency, which provide the setting in which to explore the limit of validity of the standard BG structure. It has been shown that there is a rich and intricate behavior for both the dynamics within and towards the attractors at the onset of chaos and that these two kinds of properties are linked via generalized statistical-mechanical expressions. Amongst the topics presented are: (i) permanently growing sensitivity fluctuations and their infinite family of generalized Pesin identities; (ii) the emergence of statistical-mechanical structures in the dynamics along the routes to chaos; (iii) dynamical hierarchies with modular organization; and (iv) limit distributions of sums of deterministic variables. The occurrence of generalized entropy properties in condensed-matter physical systems is illustrated by considering critical fluctuations, localization transition and glass formation. We complete our presentation with the description of the manifestations of the dynamics at the transitions to chaos in various kinds of complex systems, such as, frequency and size rank distributions and complex network images of time series. We discuss the results.