1,406 research outputs found
Entropy Monotonicity and Superstable Cycles for the Quadratic Family Revisited
The main result of this paper is a proof using real analysis of the monotonicity of the
topological entropy for the family of quadratic maps, sometimes called Milnor’s Monotonicity
Conjecture. In contrast, the existing proofs rely in one way or another on complex analysis.
Our proof is based on tools and algorithms previously developed by the authors and collaborators to
compute the topological entropy of multimodal maps. Specifically, we use the number of transverse
intersections of the map iterations with the so-called critical line. The approach is technically simple
and geometrical. The same approach is also used to briefly revisit the superstable cycles of the
quadratic maps, since both topics are closely related
Equilibrium states for potentials with \sup\phi - \inf\phi < \htop(f)
In the context of smooth interval maps, we study an inducing scheme approach
to prove existence and uniqueness of equilibrium states for potentials
with he `bounded range' condition \sup \phi - \inf \phi < \htop, first used
by Hofbauer and Keller. We compare our results to Hofbauer and Keller's use of
Perron-Frobenius operators. We demonstrate that this `bounded range' condition
on the potential is important even if the potential is H\"older continuous. We
also prove analyticity of the pressure in this context.Comment: Added Lemma 6 to deal with the disparity between leading eigenvalues
and operator norms. Added extra references and corrected some typo
The thermodynamic approach to multifractal analysis
Most results in multifractal analysis are obtained using either a
thermodynamic approach based on existence and uniqueness of equilibrium states
or a saturation approach based on some version of the specification property. A
general framework incorporating the most important multifractal spectra was
introduced by Barreira and Saussol, who used the thermodynamic approach to
establish the multifractal formalism in the uniformly hyperbolic setting,
unifying many existing results. We extend this framework to apply to a broad
class of non-uniformly hyperbolic systems, including examples with phase
transitions. In the process, we compare this thermodynamic approach with the
saturation approach and give a survey of many of the multifractal results in
the literature.Comment: 51 pages, minor corrections, added formal statements of new results
to "applications" sectio
Equilibrium states, pressure and escape for multimodal maps with holes
For a class of non-uniformly hyperbolic interval maps, we study rates of
escape with respect to conformal measures associated with a family of geometric
potentials. We establish the existence of physically relevant conditionally
invariant measures and equilibrium states and prove a relation between the rate
of escape and pressure with respect to these potentials. As a consequence, we
obtain a Bowen formula: we express the Hausdorff dimension of the set of points
which never exit through the hole in terms of the relevant pressure function.
Finally, we obtain an expression for the derivative of the escape rate in the
zero-hole limit.Comment: Minor edits. To appear in Israel J. Mat
Monotonicity of entropy for real multimodal maps
In \cite{Mil}, Milnor posed the {\em 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 \cite{MT} and for cubic maps by
Milnor & Tresser, see \cite{MTr} and also \cite{DGMT}. In this paper we will
prove the general case.Comment: Final version. To appear in Journal of the AM
- …