Invariant measures for interval maps without Lyapunov exponents
نویسندگان
چکیده
Abstract We construct an invariant measure for a piecewise analytic interval map whose Lyapunov exponent is not defined. Moreover, set of full measure, the pointwise This has Lorenz-like singularity and non-flat critical points.
منابع مشابه
Estimating Invariant Measures and Lyapunov Exponents
This paper describes a method for obtaining rigorous numerical bounds on time averages for a class of one-dimensional expanding maps. The idea is to directly estimate the absolutely continuous invariant measure for these maps, without computing trajectories. The main theoretical result is a bound on the convergence rate of the Frobenius-Perron operator for such maps. The method is applied to es...
متن کاملReturn Time Statistics for Invariant Measures for Interval Maps with Positive Lyapunov Exponent
We prove that multimodal maps with an absolutely continuous invariant measure have exponential return time statistics around a.e. point. We also show a ‘polynomial Gibbs property’ for these systems, and that the convergence to the entropy in the Ornstein-Weiss formula has normal fluctuations. These results are also proved for equilibrium states of some Hölder potentials.
متن کاملOn the estimation of invariant measures and Lyapunov exponents arising from iid compositions of maps
We present a method of approximating the unique invariant measure and associated Lyapunov exponents of a random dynamical system deened by the iid composition of a family of maps in situations where only one Lyapunov exponent is observed. As a corollary, our construction also provides a method of estimating the top Lyapunov exponent of an iid random matrix product. We develop rigorous numerical...
متن کاملReturn Time Statistics of Invariant Measures for Interval Maps with Positive Lyapunov Exponent
We prove that multimodal maps with an absolutely continuous invariant measure have exponential return time statistics around almost every point. We also show a ‘polynomial Gibbs property’ for these systems, and that the convergence to the entropy in the Ornstein-Weiss formula has normal fluctuations. These results are also proved for equilibrium states of some Hölder potentials.
متن کاملInvariant Measures for Interval Maps with Critical Points and Singularities
We prove that, under a mild summability condition on the growth of the derivative on critical orbits any piecewise monotone interval map possibly containing discontinuities and singularities with infinite derivative (cusp map) admits an ergodic invariant probability measures which is absolutely continuous with respect to Lebesgue measure.
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
ژورنال
عنوان ژورنال: Ergodic Theory and Dynamical Systems
سال: 2021
ISSN: ['0143-3857', '1469-4417']
DOI: https://doi.org/10.1017/etds.2021.128