This Package provides methods for the rigorous approximation of Absolutely Continuous Invariant Measures for one dimensional dynamical systems, using the results in [2] and subsequent [1].
By Birkhoff Ergodic Theorem we know that if a dynamical system T admits an ergodic invariant measure μ, for μ-almost every point x the frequence of the visits to a set E corresponds to the measure of the set with respect to μ.
Therefore, being able to approximate invariant measures with a large basin is interesting to investigate the statistical properties of the dynamical system T. [5]
The existence of absolutely continuous invariant measures for one dimensional maps is a delicate topic; with this package we present approximation schemes for the invariant measures of system that satisfy a Lasota-Yorke inequality through the use of a coarse-fine scheme and a posteriori estimates on the mixing time, i.e., this means that the mixing time is estimated by our algorithm and we do not need an a priori estimate, which is usually difficult to obtain.
The Ulam approximation schemes works under relatively weak hypothesis on the dynamics and was used in [4] to approximate the invariant measure for the geometric Lorenz 1-dimensional map.
We are currently working on the implementation of the Ulam scheme for system with additive uniform noise, as the one used in [2]
Examples of usage are present in the directory examples.
using InvariantMeasures
D = Mod1Dynamic(x -> 4x + 0.01InvariantMeasures.sinpi(8x))
B = Ulam(1024)
Q = DiscretizedOperator(B, D)The code snippet above defines a dynamic obtained by reducing f(x) = 4x+0.01 sin(8πx) modulo 1, a basis B associated to the Ulam discretization on a partition of 1024 homogenous intervals, and computes the discretized operator Q, a Markov chain whose entries are P[T(x)∈ Iᵢ | x ∈ Iⱼ].
Note the usage of InvariantMeasures.sinpi(8*x) rather than Base.sinpi or Base.sin(8\pi*x). This detail is required to ensure that f(1) == 4 exactly.
norms = powernormbounds(B, D; Q=Q)This function computes the L¹ norm of Q^k, for k = 1,2,...,k_max (up to a sufficiently large number of powers to observe decay) when restricted to the space U of average-0 vectors. This gives us the a posteriori estimate for the mixing time of the Markov chain and is used in our rigorous estimate.
w = invariant_vector(B, Q)
distance_from_invariant(B, D, Q, w, norms)This computes a (non-rigorous) approximation of the invariant measure of D; then distance_from_invariant computes an upper bound for the L¹ distance between w and the density of the absolutely continuous invariant measure of the system.
Inside the examples it is showed how to use the coarse-fine scheme to obtain better L¹ bounds and reduce the computational time.
The function sinpi in the interval arithmetic package that we are using relies on the CRlibm.jl package, which currently works only under Linux. So the examples that use trigonometric functions only work on this OS.
[1] Galatolo S., Monge M., Nisoli I., Poloni F. A general framework for the rigorous computation of invariant densities and the coarse-fine strategy https://doi.org/10.48550/arXiv.2212.05017
[2] Galatolo S., Monge M., Nisoli I., Existence of noise induced order, a computer aided proof Nonlinearity 33 (9), 4237 (2020)
[3] Galatolo S., Nisoli I., An elementary approach to rigorous approximation of invariant measures SIAM J. Appl Dyn Sys.13 pp. 958-985 (2014)
[4] Galatolo S., Nisoli I. Rigorous computation of invariant measures and fractal dimension for maps with contracting fibers: 2D Lorenz-like maps
Ergodic Theory and Dynamical Systems 36 (6), 1865-1891 (2016)
[5] Viana M., Olivera K. Foundations of Ergodic Theory Cambridge studies in advanced mathematics, Cambridge University Press 2016