Applications of Variable Discounting Dynamic Programming to Iterated Function Systems and Related Problems

TL;DR

This paper investigates fixed point solutions of non-linear variable discounted transfer operators in dynamic programming, establishing their properties and applying these results to various complex systems and optimization problems.

Contribution

It introduces a unified approach to analyze fixed points of variable discounted operators and applies it to multiple areas including iterated function systems and ergodic optimization.

Findings

Proved existence and uniqueness of fixed points for a broad class of operators.

Established regularity properties with respect to discount and immediate return.

Reformulated key variational problems in dynamic programming context.

Abstract

We study existence and uniqueness of the fixed points solutions of a large class of non-linear variable discounted transfer operators associated to a sequential decision-making process. We establish regularity properties of these solutions, with respect to the immediate return and the variable discount. In addition, we apply our methods to reformulating and solving, in the setting of dynamic programming, some central variational problems on the theory of iterated function systems, Markov decision processes, discrete Aubry-Mather theory, Sinai-Ruelle-Bowen measures, fat solenoidal attractors, and ergodic optimization.