Generic Hyperbolicity in the Logistic Family

Extremes and Recurrence in Dynamical Systems

“…point in [0, 1] is also attracted to that cycle and there is no absolutely continuous invariant measure. The set of λ for which f λ is Axiom A is open and dense in [0, 4] ( [189,190]). We focus on the case of the positive Lebesgue measure set of parameters λ such that there is an absolutely continuous invariant measure for f λ [191].…”

The Work of the 2014 Fields Medalists

“…These different dynamical behaviors were already present in the famous quadratic family x ∈ [0, 1] q µ = µx (1 − x), where the parameter µ belongs to the interval (0, 4]. The set of parameter values corresponding to regular maps is open and dense by a difficult result proved in [18] and [20]. On the other hand, the set of parameter values corresponding to stochastic maps has positive Lebesgue measure, as proved in [20], and the complement has measure zero [22] but positive Hausdorff dimension.…”

“…It is a nuisance when counting degree 4 fields, for instance, because each field contains many orders (subrings that are free over Z of the same rank as the field), and each order can have many cubic resolvent rings. But it is a boon if the extra structure itself is worth counting; for instance, one of the prehomogeneous spaces enabled Bhargava [2, Theorem 5] to prove the first nontrivial case of the Cohen-Martinet generalization [18] of the Cohen-Lenstra heuristics on the distribution of class groups.…”

Decidability of Chaos for Some Families of Dynamical Systems

“…Related to this notion we have the following theorem due to Graczyk and Swiatek [8] and Lyubich [10].…”