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We develop algorithms and techniques to compute rigorous bounds for finite pieces of orbits of the critical points, for intervals of parameter values, in the quadratic family of one-dimensional maps f a (x) = a − x 2. We illustrate the effectiveness of our approach by constructing a dynamically defined partition P of the parameter interval Ω = [ 1.4 , 2 ] into almost 4 × 10 6 subintervals, for each of which we compute to high precision the orbits of the critical points up to some time N and other dynamically relevant quantities, several of which can vary greatly, possibly spanning several orders of magnitude. We also subdivide P into a family P + of intervals, which we call stochastic intervals, and a family P − of intervals, which we call regular intervals. We numerically prove that each interval ω ∈ P + has an escape time, which roughly means that some iterate of the critical point taken over all the parameters in ω has considerable width in the phase space. This suggests, in turn, that most parameters belonging to the intervals in P + are stochastic and most parameters belonging to the intervals in P − are regular, thus the names. We prove that the intervals in P + occupy almost 90% of the total measure of Ω. The software and the data are freely available at http://www.pawelpilarczyk.com/quadr/, and a web page is provided for carrying out the calculations. The ideas and procedures can be easily generalized to apply to other parameterized families of dynamical systems. [ABSTRACT FROM AUTHOR]

We formulate a notion of (uniform) asymptotic involutivity and show that it implies (unique) integrability of corank-1 continuous distributions in dimensions three or less. This generalizes and extends a classical Frobenius theorem, which says that an involutive distribution is uniquely integrable. [ABSTRACT FROM AUTHOR]

We prove that a class of one-dimensional maps with an arbitrary number of non-degenerate critical and singular points admits an induced Markov tower with exponential return time asymptotics. In particular the map has an absolutely continuous invariant probability measure with exponential decay of correlations for Hölder observations. [ABSTRACT FROM AUTHOR]