Sebastian van Strien

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.

LOCAL CONNECTIVITY OF THE JULIA SET OF REAL POLYNOMIALS

In particular, the Julia set of z + c1 is locally connected if c1 ∈ [−2, 1/4] and totally disconnected if c1 ∈ R \ [−2, 1/4] (note that [−2, 1/4] is equal to the set of parameters c1 ∈ R for which the critical point c = 0 does not escape to infinity). This answers a question posed by Milnor, see [Mil1]. We should emphasize that if the ω-limit set ω(c) of the critical point c = 0 is not minimal then it very easy to see that the Julia set is locally connected, see for example Section 10. Yoccoz [Y] already had shown that each quadratic polynomial which is only finitely often renormalizable (with non-escaping critical point and no neutral periodic point) has a locally connected Julia set. Moreover, Douady and Hubbard [DH1] already had shown before that each polynomial of the form z 7→ z + c1 with an attracting or neutral parabolic cycle has a locally connected Julia set. As will become clear, the difficult case is the infinitely renormalizable case. In fact, using the reduction method developed in Section 3 of this paper, it turns out that in the non-renormalizable case the Main Theorem follows from some results in [Ly3] and [Ly5], see the final section of this paper.

Invariant measures exist under a summability condition for unimodal maps

SummaryFor unimodal maps with negative Schwarzian derivative a sufficient condition for the existence of an invariant measure, absolutely continuous with respect to Lebesgue measure, is given. Namely the derivatives of the iterations of the map in the (unique) critical value must be so large that the sum of (some root of) the inverses is finite.

Local connectivity and quasi-conformal rigidity of non-renormalizable polynomials

We prove that topologically conjugate non-renormalizable polynomials are quasi-conformally conjugate. From this we derive that each such polynomial can be approximated by a hyperbolic polynomial. As a by-product we prove that the Julia set of a non-renormalizable polynomial with only hyperbolic periodic points is locally connected, and the Branner-Hubbard conjecture. The main tools are the enhanced nest construction (developed in a previous joint paper with [Rigidity for real polynomials, Ann. of Math. (2) 165 (2007) 749-841.]) and a lemma of Kahn and Lyubich (for which we give an elementary proof in the real case).

Smooth linearization of hyperbolic fixed points without resonance conditions

A well known theorem of Hartman-Grobman says that a C1 diffeomorphism f:Rn→Rn with a hyperbolic fixed point at 0 can be conjugated to the linear diffeomorphism L = df(0) (at least in a neighbourhood of 0). In this paper we will show that if ƒ is C2 then ƒ is differentiably conjugate to L at 0; moreover, the conjugacy is Holder outside 0. No resonance conditions will be required.

Fictitious play in 3×3 games: The transition between periodic and chaotic behaviour

In the 1960s Shapley provided an example of a two-player fictitious game with periodic behaviour. In this game, player A aims to copy Bs behaviour and player B aims to play one ahead of player A. In this paper we generalise Shapleys example by introducing an external parameter. We show that the periodic behaviour in Shapleys example at some critical parameter value disintegrates into unpredictable (chaotic) behaviour, with players dithering a huge number of times between different strategies. At a further critical parameter the dynamics becomes periodic again, but now both players aim to play one ahead of the other. In this paper we adopt a geometric (dynamical systems) approach. Here we prove rigorous results on continuity of the dynamics and on the periodic behaviour, while in the sequel to this paper we shall describe the chaotic behaviour.

On Cherry flows

The purpose of this research is to describe all smooth vector fields on the torus T 2 with a finite number of singularities, no periodic orbits and no saddleconnections. In this paper we are able to complete the description within the class of vector fields which are area contracting near all singularities. In particular we give a large class of analytic vector fields on the torus T 2 which have non-trivial recurrence and also sinks.

Monotonicity of entropy for real multimodal maps

In [16], Milnor posed the Monotonicity Conjecture that the set of parameters within a family of real multimodal polynomial interval maps, for which the topological entropy is constant, is connected. This conjecture was proved for quadratic by Milnor & Thurston [17] and for cubic maps by Milnor & Tresser, see [18] and also [5]. In this paper we will prove the general case.

Singularities of vector fields on 3 determined by their first non-vanishing jet

In this paper we will present a result which gives a sufficient condition for a vector field X on ℝ 3 to be equivalent at a singularity to the first non-vanishing jet j k X( p ) of X at p . This condition - which only depends on the homogeneous vector field defined by j k X ( p ) - is stated in terms of the blown-up vector field (which is defined on S 2 xℝ), and essentially means that there are no saddleconnections for | S 2 ×{0}. The key tool in the proof will be a result of local normal linearization along a codimension 1 submanifold M providing a C 0 conjugacy having a normal derivative along M equal to 1.

Hyperbolicity properties of ² multi-modal Collet-Eckmann maps without Schwarzian derivative assumptions

In this paper we study the dynamical properties of general C2 maps f: [0, 1] -+ [0, 1] with quadratic critical points (and not necessarily unimodal). We will show that if such maps satisfy the well-known Collet-Eckmann conditions then one has (a) hyperbolicity on the set of periodic points; (b) nonexistence of wandering intervals; (c) sensitivity on initial conditions; and (d) exponential decay of branches (intervals of monotonicity) of fn as n oo; For these results we will not make any assumptions on the Schwarzian derivative f. We will also give an estimate of the return-time of points that start near critical points.