1 Introduction
1.1 Some Motivating Applications of IFS
1.2 Organization of the Survey and Past Surveys on IFS
2 Introduction and Historical Background of IFS
2.1 Mathematical Preliminaries on IFS
3 IFS on Compact State Space
3.1 State Independent IFS with Finite Number of Self Maps
3.2 Ergodic Properties of the Associated Markov Processes
3.3 State-Dependent IFS with Finite Number of Self Maps
3.4 Ergodic Properties of the Associated Markov Processes
3.5 State-Dependent IFS with Infinite Number of Maps
4 IFS on Polish Spaces
4.1 State-Independent IFS with Infinite Number of Self Maps
5 Ergodic Theorems and Law of Large Numbers for IFS
5.1 A Motivating Example from Deterministic Dynamics
5.2 A Motivating Example from Random Dynamics
5.3 Ergodic Theorems for IFS on Compact Space
5.4 Ergodic Theorems for IFS on Polish Space
6 Central Limit Theorem (CLT) and Invariance Principle for IFS
6.1 CLT for IFS with State-Independent Probabilities with Countably Infinite Number of Self Maps on Polish Space
6.2 Invariance Principle of IFS
7 Geometry of the Unique Invariant Measure
7.1 A Special Case: I.I.D Random Iteration of Quadratic and Logistic Maps
7.2 Decomposition of Measure for IFS
7.3 Topological Aspects of IFS
7.4 Conformal IFS and Teichmüller Theory for IFS
8 Opportunities for Further Research
8.1 Non-asymptotic Study of Mixing Rates in IFS
8.2 Continuous-Time IFS and IFS on Time Scales
8.3 IFS with Discontinuous Probability Functions
8.4 Testable Conditions for the Existence and Non-Existence of a Unique Invariant Measure
8.5 Approximating and Shaping of Invariant Measure for IFS
8.6 Set-valued IFS
8.7 Non-homogeneous IFS
8.8 Large Deviations for IFS
9 In Place of Conclusions