This monograph is devoted to developing a theory of combined measure and shift invariance of time scales with the related applications to shift functions and dynamic equations. The study of shift closeness of time scales is significant to investigate the shift functions such as the periodic functions, the almost periodic functions, the almost automorphic functions, and their generalizations with many relevant applications in dynamic equations on arbitrary time scales.
First proposed by S. Hilger, the time scale theorya unified view of continuous and discrete analysishas been widely used to study various classes of dynamic equations and models in real-world applications. Measure theory based on time scales, in its turn, is of great power in analyzing functions on time scales or hybrid domains.
As a new and exciting type of mathematicsand more comprehensive and versatile than the traditional theories of differential and difference equations, the time scale theory can precisely depict the continuous-discrete hybrid processes and is an optimal way forward for accurate mathematical modeling in applied sciences such as physics, chemical technology, population dynamics, biotechnology, and economics and social sciences.
Graduate students and researchers specializing in general dynamic equations on time scales can benefit from this work, fostering interest and further research in the field. It can also serve as reference material for undergraduates interested in dynamic equations on time scales. Prerequisites include familiarity with functional analysis, measure theory, and ordinary differential equations.

Develops a theory of combined measure and shift-invariance of time scales Illustrates with relevant applications to shift functions and dynamic equations Emphasizes the power of this theory for accurate mathematical modeling in applied sciences

Autorentext
Chao Wang is a Professor and PhD in Mathematics at Yunnan University in China. Dr. Wang has authored the book "Theory of Translation Closedness for Time Scales" (978-3-030-38643-6), published by Springer. His research focuses on the fields of nonlinear dynamic systems, control theory, fuzzy dynamic equations, fractional differential equations, bifurcation theory, nonlinear analysis, and numerical modeling.

**Ravi P. Agarwal** is a Professor at the Texas A&M University-Kingsville, USA. He completed his PhD at the Indian Institute of Technology, Madras, India, in 1973. Dr. Agarwal has published 1700 research articles in several different fields and authored or co-authored 50 books, including "Theory of Translation Closedness for Time Scales" (978-3-030-38643-6), published by Springer.

Inhalt
Riemann Integration, Stochastic Calculus and Shift Operators on Time Scales.- -Measurability and Combined Measure Theory on Time Scales .- Shift Invariance and Matched Spaces of Time Scales.- Almost Periodic Functions under Matched Spaces of Time Scales.- Almost Automorphic Functions under Matched Spaces of Time Scales.- C0-Semigroup and Stepanov-like Almost Automorphic Functions on Hybrid Time Scales.- Almost Periodic Dynamic Equations under Matched Spaces.- Almost Automorphic Dynamic Equations under Matched Spaces.- Applications on Dynamics Models under Matched Spaces.