1. Background:

The aim of ergodic theory is to understand the stochastic behavior of deterministic dynamical systems by studying the ergodic invariant probability measures of the system. For the study of ergodic theory on symbolic dynamical systems, we intend to introduce some recently developed topics in dynamical systems. In the case of audience who does not acquire the background needed, we start with considering some topological behavior and complexity of one-dimensional dynamical systems. Beginning at the conjugacy and graph representation of systems, an important invariant known as the topological entropy is introduced. Followed by the decidability of conjugacy between two systems, an application of the topological entropy, the existence of embedding or factor map between given systems, is introduced.

2. Outline:  

Lecturer: Prof. Chih-Hung Chang(chchang@nuk.edu.tw)

Department of Applied Mathematics, National University of Kaohsiung

Date:Feb. 22, 2021 - June 21, 2021

Title:

1. Shift Spaces
2. Higher Block Shifts and Higher Power Shifts
3. Sliding Block Codes
4. Shifts of finite type
5. Graph Representations of Shifts of Finite Type
6. State Splitting
7. Sofic Shifts
8. Characterizations of Sofic Shifts
9. Minimal Right-Resolving Presentations
10. Entropy
11. Perron-Frobenius Theory
12. Irreducible Components and Cyclic Structure
13. Shifts as Dynamical Systems
14. Invariants and Zeta Functions
15. Markov Partitions

3. Credit: 3