Edson Vargas

Real bounds, ergodicity and negative Schwarzian for multimodal maps

Over the last 20 years, many of the most spectacular results in the field of dynamical systems dealt specifically with interval and circle maps (or perturbations and complex extensions of such maps). Primarily, this is because in the one-dimensional case, much better distortion control can be obtained than for general dynamical systems. However, many of these spectacular results were obtained so far only for unimodal maps. The aim of this paper is to provide all the tools for studying general multimodal maps of an interval or a circle, by obtaining * real bounds controlling the geometry of domains of certain first return maps, and providing a new (and we believe much simpler) proof of absense of wandering intervals; * provided certain combinatorial conditions are satisfied, large real bounds implying that certain first return maps are almost linear; * Koebe distortion controlling the distortion of high iterates of the map, and negative Schwarzian derivative for certain return maps (showing that the usual assumption of negative Schwarzian derivative is unnecessary); * control of distortion of certain first return maps; * ergodic properties such as sharp bounds for the number of ergodic components.

Entropy of flows, revisited

We introduce a concept of measure-theoretic entropy for flows and study its invariance under measure-theoretic equivalences. Invariance properties of the corresponding topological entropy is studied too. We also answer a question posed by Bowen-Walters in [3] concerning the equality between the topological entropy of the time-one map of an expansive flow and the time-one map of its symbolic suspension.

Non-trivial wandering domains and homoclinic bifurcations

We prove that on any surface there is a C ∞ diffeomorphism exhibiting a wandering domain D with the following ergodic property: for any orbit starting in D the corresponding Birkhoff mean of Dirac measures converges to the invariant measure supported on a hyperbolic horseshoewhich is equivalent to the unique non-trivial Hausdorff measure in � . The construction is obtained by perturbation of a diffeomorphism such that the unstable and stable foliations of this horseshoeare relatively thick and in tangential position. We describe, in addition, the set of accumulation points of orbits starting in D.

Invariant Measures for Cherry Flows

We investigate the invariant probability measures for Cherry flows, i.e. flows on the two-torus which have a saddle, a source, and no other fixed points, closed orbits or homoclinic orbits. In the case when the saddle is dissipative or conservative we show that the only invariant probability measures are the Dirac measures at the two fixed points, and the Dirac measure at the saddle is the physical measure. In the other case we prove that there exists also an invariant probability measure supported on the quasi-minimal set, we discuss some situations when this other invariant measure is the physical measure, and conjecture that this is always the case. The main techniques used are the study of the integrability of the return time with respect to the invariant measure of the return map to a closed transversal to the flow, and the study of the close returns near the saddle.

Markov partition in non-hyperbolic interval dynamics

We considerC2 unimodal mapsf such that all periodic points are hyperbolic, the critical point is non-degenerated and non-recurrent, and the Julia set does not contain intervals. We construct a Markov partition for a big part of the Julia set. Then we use it to estimate the limit capacity and Hausdorff dimension of the Julia set.

A full family of multimodal maps on the circle

We exhibit a family of trigonometric polynomials inducing a family of 2m- multimodal maps on the circle which contains all relevant dynamical behavior.

Erratum to “Real bounds, ergodicity and negative Schwarzian for multimodal maps”

Here, as before, we define an open interval K to be nice if no iterate of ∂K enters K. This implies that if K1 and K2 are pullbacks of K, then they are either disjoint or nested. In Lemma 9 (page 762) it was implicitly assumed that V is disjoint from Jn. It is for this reason that the proof of Theorem C(1) does not work unless we assume V is nice (or something similar). The proof of Theorem C(1) as stated above is essentially the same as before, using Lemma 6′ below instead of Lemma 6; then in Lemma 9 (page 762) we do not need to require that kn+1 is a jump time provided we assume that V is nice. Making the additional assumption that V is nice, Proposition 1 (and its proof) and the rest of the paper go through unchanged. Lemma 6′. For each ρ > 0 sufficiently small, there exists δ3 > 0 such that if I is a ρ-scaled neighbourhood of a nice interval V ⊂ I, then J is a δ3-scaled neighbourhood of any component A of φ−k |J (V ) (where k ≥ 1 is arbitrary).

Bifurcation frequency for unimodal maps

We consider some natural one-parameter unfoldingsfμ, of a unimodal mapf0 whose periodic points are hyperbolic and whose critical point is nondegenerate and eventually periodic. Among other facts, it follows from our theorems that, if the Julia set off0 does not contain intervals, the relative measure of the bifurcation set is zero at zero.