Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userStatistical properties of one-dimensional maps with critical points and singularities
156
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
We prove that a class of one-dimensional maps with an arbitrary number of non-degenerate critical and singular points admits an induced Markov tower with exponential return time asymptotics. In particular the map has an absolutely continuous invariant probability measure with exponential decay of correlations for H{o}lder observations.
rate research
Read More
For an infinitely renormalizable negative Schwarzian unimodal map $f$ with non-flat critical point, we analyze statistical properties of periodic points as the periods tend to infinity. Introducing a weight function $varphi$ which is a continuous or a geometric potential $varphi=-betalog|f|$ ($betainmathbb R$), we establish the level-2 Large Deviation Principle for weighted periodic points. From this, we deduce that all weighted periodic points equidistribute with respect to equilibrium states for the potential $varphi$. In particular, it follows that all periodic points are equidistributed with respect to measures of maximal entropy, and all periodic points weighted with their Lyapunov exponents are equidistributed with respect to the post-critical measure supported on the attracting Cantor set.
Given a piecewise $C^{1+beta}$ map of the interval, possibly with critical points and discontinuities, we construct a symbolic model for invariant probability measures with nonuniform expansion that do not approach the critical points and discontinuities exponentially fast almost surely. More specifically, we code the lift of these measures in the natural extension of the map.
In this paper we investigate how many periodic attractors maps in a small neighbourhood of a given map can have. For this purpose we develop new tools which help to make uniform cross-ratio distortion estimates in a neighbourhood of a map with degenerate critical points.
We construct Markov partitions for non-invertible and/or singular nonuniformly hyperbolic systems defined on higher dimensional Riemannian manifolds. The generality of the setup covers classical examples not treated so far, such as geodesic flows in closed manifolds, multidimensional billiard maps, and Viana maps, and includes all the recent results of the literature. We also provide a wealth of applications.
Continuum models with critical end points are considered whose Hamiltonian ${mathcal{H}}[phi,psi]$ depends on two densities $phi$ and $psi$. Field-theoretic methods are used to show the equivalence of the critical behavior on the critical line and at the critical end point and to give a systematic derivation of critical-end-point singularities like the thermal singularity $sim|{t}|^{2-alpha}$ of the spectator-phase boundary and the coexistence singularities $sim |{t}|^{1-alpha}$ or $sim|{t}|^{beta}$ of the secondary density $<psi>$. The appearance of a discontinuity eigenexponent associated with the critical end point is confirmed, and the mechanism by which it arises in field theory is clarified.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
