Abstract. Given a one-parameter family f~.(x) of maps of the interval [0, 1], we consider the set of parameter values 2 for which f~ has an invariant measure absolutely continuous with respect to Lebesgue measure. We show that this set has positive measure, for two classes of maps: i) fx(x)= 2f(x) where 0 < 2 < 4 and f(x) is a function C3-near the quadratic map x(1 -x), and ii) f~(x)=2f(x) (mod 1) where f is C a, f(0) =f(1) = 0 and f has a unique nondegenerate critical point in [0, t].

It is shown that for certain classes of S-unimodal maps with aperiodic kneading sequences, the topological conjugacies are also quasisymmetric. This includes some infinitely renormalizable polynomials of unbounded type.

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For an arbitrary non-renormalizable unimodal map of the interval,f:I→I, with negative Schwarzian derivative, we construct a related mapFdefined on a countable union of intervals Δ. For each interval Δ,Frestricted to Δ is a diffeomorphism which coincides with some iterate offand whose range is a fixed subinterval ofI. IfFsatisfies conditions of the Folklore Theorem, we callfexpansion inducing. Letcbe a critical point off. Forfsatisfyingf″(c) ≠ 0, we give sufficient conditions for expansion inducing. One of the consequences of expansion inducing is that Milnor's conjecture holds forf: the ω-limit set of Lebesgue almost every point is the interval [f2,f(c)]. An important step in the proof is a starting condition in the box case: if for initial boxes the ratio of their sizes is small enough, then subsequent ratios decrease at least exponentially fast and expansion inducing follows.

We formulate some su cient conditions for the existence of Sinai-Ruelle-Bowen measures for piecewise C 2 di eomorphisms with unbounded derivatives. The result can be viewed as a two-dimensional version of the well known one-dimensional Folklore Theorem on the existence of absolutely continuous invariant measures. Here we formulate the results and outline the main ideas and tools of our approach. The detailed version will appear elsewhere.