Decay of geometry for unimodal maps: An elementary proof

Abstract

We prove that a nonrenormalizable smooth unimodal interval map with critical order between 1 and 2 displays decay of geometry, by an elementary and purely "real" argument. This completes a "real" approach to Milnor's attractor problem for smooth unimodal maps with critical order not greater than 2.

Existence of unique SRB-measures is typical for real unicritical polynomial families

“…Let DG denote the collection of parameters c for which f c satisfies the decaying geometry condition. We should note that if = 2, F 0 r ⊂ DG (and in fact, the decay is at least exponentially fast, see [14,21,35]). We will first deal with the parameters c ∈ F 0 r \ DG, and show that f c |ω(0) is uniquely ergodic, and there is a unique SRB-measure which is either an acip or the invariant probability measure supported on ω(0).…”

Local Critical Perturbations of Unimodal Maps

“…Finally we state a result which we rely upon considering the finitely renormalizable recurrent case in Section 4. It has been proven in [2] and [11], and establishes the decay of geometry for maps in U 3 . Moreover, the same result is true in U 2 , see [11] and [12].…”

“…In Section 4 we consider finitely renormalizable unimodal maps in U 2 and prove that if they are not stabilized then they are not c-structurally stable. The decay of geometry established in [2] and [11] is crucial for our arguments. Then in Section 5 we obtain similar results for infinitely renormalizable maps in U 2 (and even in U ∞ ).…”

Solenoidal attractors with bounded combinatorics are shy

“…We say that the cocycle L is uniformly expanding if there exist K > 0 and θ > 1 such that for every (x, v) ∈ E we have (38) |v i | ≥ Kθ i |v| Proof. Of course if L : E → E is uniformly expanding then (39) holds.…”