On the structure of isentropes of polynomial maps

The structure of isentropes (i.e. level sets of constant topological entropy) including the monotonicity of entropy, has been studied for polynomial interval maps since the 1980s. We show that isentropes of multimodal polynomial families need not be locally connected and that entropy does in general not depend monotonically on a single critical value.Comment: 16 page

Payoff Performance of Fictitious Play

We investigate how well continuous-time fictitious play in two-player games performs in terms of average payoff, particularly compared to Nash equilibrium payoff. We show that in many games, fictitious play outperforms Nash equilibrium on average or even at all times, and moreover that any game is linearly equivalent to one in which this is the case. Conversely, we provide conditions under which Nash equilibrium payoff dominates fictitious play payoff. A key step in our analysis is to show that fictitious play dynamics asymptotically converges the set of coarse correlated equilibria (a fact which is implicit in the literature).Comment: 16 pages, 4 figure

Density of hyperbolicity for classes of real transcendental entire functions and circle maps

We prove density of hyperbolicity in spaces of (i) real transcendental entire functions, bounded on the real line, whose singular set is finite and real and (ii) transcendental self-maps of the punctured plane which preserve the circle and whose singular set (apart from zero and infinity) is contained in the circle. In particular, we prove density of hyperbolicity in the famous Arnol'd family of circle maps and its generalizations, and solve a number of other open problems for these functions, including three conjectures by de Melo, Salom\~ao and Vargas. We also prove density of (real) hyperbolicity for certain families as in (i) but without the boundedness condition. Our results apply, in particular, when the functions in question have only finitely many critical points and asymptotic singularities, or when there are no asymptotic values and the degree of critical points is uniformly bounded.Comment: 46 pages, 3 figures. V5: Final peer-reviewed accepted manuscript, to appear in Duke Mathematical Journal. Only minor changes from the previous (significantly revised) version V

Decay of correlations in one-dimensional dynamics

We consider multimodal C^3 interval maps f satisfying a summability condition on the derivatives D_n along the critical orbits which implies the existence of an absolutely continuous f -invariant probability measure mu. If f is non-renormalizable, mu is mixing and we show that the speed of mixing (decay of correlations) is strongly related to the rate of growth of the sequence D_n as n tends to infinity . We also give sufficient conditions for mu to satisfy the Central Limit Theorem. This applies for example to the quadratic Fibonacci map which is shown to have subexponential decay of correlations.Comment: To appear in Annales de l'Ecole Normale Superieure, 200

Absorbing Cantor sets in dynamical systems: Fibonacci maps

In this paper we shall show that there exists a polynomial unimodal map f: [0,1] -> [0,1] which is 1) non-renormalizable(therefore for each x from a residual set, $\omega(x)$ is equal to an interval), 2) for which $\omega(c)$ is a Cantor set, and 3) for which $\omega(x)=\omega(c)$ for Lebesgue almost all x. So the topological and the metric attractor of such a map do not coincide. This gives the answer to a question posed by Milnor

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