26 research outputs found
Natural equilibrium states for multimodal maps
This paper is devoted to the study of the thermodynamic formalism for a class
of real multimodal maps. This class contains, but it is larger than,
Collet-Eckmann. For a map in this class, we prove existence and uniqueness of
equilibrium states for the geometric potentials , for the largest
possible interval of parameters . We also study the regularity and convexity
properties of the pressure function, completely characterising the first order
phase transitions. Results concerning the existence of absolutely continuous
invariant measures with respect to the Lebesgue measure are also obtained
Quotient Complexity of Regular Languages
The past research on the state complexity of operations on regular languages
is examined, and a new approach based on an old method (derivatives of regular
expressions) is presented. Since state complexity is a property of a language,
it is appropriate to define it in formal-language terms as the number of
distinct quotients of the language, and to call it "quotient complexity". The
problem of finding the quotient complexity of a language f(K,L) is considered,
where K and L are regular languages and f is a regular operation, for example,
union or concatenation. Since quotients can be represented by derivatives, one
can find a formula for the typical quotient of f(K,L) in terms of the quotients
of K and L. To obtain an upper bound on the number of quotients of f(K,L) all
one has to do is count how many such quotients are possible, and this makes
automaton constructions unnecessary. The advantages of this point of view are
illustrated by many examples. Moreover, new general observations are presented
to help in the estimation of the upper bounds on quotient complexity of regular
operations
The full renormalization horseshoe for unimodal maps of higher degree: exponential contraction along hybrid classes
We prove exponential contraction of renormalization along hybrid classes of
infinitely renormalizable unimodal maps (with arbitrary combinatorics), in any
even degree . We then conclude that orbits of renormalization are asymptotic
to the full renormalization horseshoe, which we construct. Our argument for
exponential contraction is based on a precompactness property of the
renormalization operator ("beau bounds"), which is leveraged in the abstract
analysis of holomorphic iteration. Besides greater generality, it yields a
unified approach to all combinatorics and degrees: there is no need to account
for the varied geometric details of the dynamics, which were the typical source
of contraction in previous restricted proofs.Comment: 44 page
Power spectra for deterministic chaotic dynamical systems
We present results on the broadband nature of power spectra for large classes of discrete chaotic dynamical systems, including uniformly hyperbolic (Axiom A) diffeomorphisms and certain nonuniformly hyperbolic diffeomorphisms (such as the Henon map). Our results also apply to noninvertible maps, including Collet-Eckmann maps. For such maps (even the nonmixing ones) and Holder continuous observables, we prove that the power spectrum is analytic except for finitely many removable singularities, and that for typical observables the spectrum is nowhere zero. Indeed, we show that the power spectrum is bounded away from zero except for infinitely degenerate observables. For slowly mixing systems such as Pomeau-Manneville intermittency maps, where the power spectrum is at most finitely differentiable, nonvanishing of the spectrum remains valid provided the decay of correlations is summable