3,694 research outputs found
Milnor’s Conjecture on Monotonicity of Topological Entropy: results and questions
This note discusses Milnor’s conjecture on monotonicity of entropy and gives a short exposition of the ideas used in its proof which was obtained in joint work with Henk Bruin, see [BvS09]. At the end of this note we explore some related conjectures and questions
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
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