Communications in Mathematical Sciences

Volume 16 (2018)

Number 8

A quadratic spline least squares method for computing absolutely continuous invariant measures

Pages: 2077 – 2093



Duanmei Zhou (College of Mathematics and Computer Science, Gannan Normal University, Ganzhou, China)

Guoliang Chen (Department of Mathematics, East China Normal University, Shanghai, China)

Jiu Ding (Department of Mathematics, University of Southern Mississippi, Hattiesburg, Miss., U.S.A.)

Noah H. Rhee (Department of Mathematics and Statistics, University of Missouri, Kansas City, Mo., U.S.A.)


We develop a quadratic spline approximation method for the computation of absolutely continuous invariant measures of one dimensional mappings, based on the orthogonal projection of $L^2$ spaces. We prove the norm convergence of the numerical scheme and present the numerical experiments.


Frobenius–Perron operators, invariant measures, least squares approximations, spline functions

2010 Mathematics Subject Classification

37M25, 65J10, 65P20

The work of D. Zhou was supported by the National Natural Science Foundation of China (Nos. 11861008, 11501126, 61502107, 11661007, 11661008, 61863001); the research fund of Gannan Normal University (No. 18zb04); the key disciplines coordinate innovation projects of Gannan Normal University; and the Support of the Development for Local Colleges and Universities Foundation of China - Applied Mathematics Innovative Team Building.

The work of G. Chen was supported by the National Natural Science Foundation of China (No. 11471122).

The work of J. Ding was supported by the 111 Project and the Shanghai Key Laboratory of Pure Mathematics and Mathematical Practice.

Received 6 December 2017

Accepted 13 August 2018

Published 18 April 2019