Abstract
The paper aims to deduce the relation between the category of topology and algebra from viewpoint of geometry and dynamical system. We introduce and define a dynamical manifold as a manifold associated with a time parameter. We obtain the induced chain of topological dynamics on the fundamental group from the chain of dynamical maps on a dynamical manifold. For many adjunctions in this context, we deduce the limit topological dynamics and conditional topological dynamics on the fundamental group. We use the category of commutative diagrams as chains of dynamical manifolds to deduce the chains on fundamental groups. Also, we describe how the manifold changes in a dynamical system from the view of the fundamental group.
Highlights
Introduction and DefinitionsA dynamical system might be an arrangement that grows in time via the iterative application of a dynamical rule. e evolution rule shows the transformation of the real state under its conditions as well as possibly earlier conditions.e fact that state evolutions must be based on the system’s states suggests that the dynamics are recursive
Input does not need to be given to the system continuously, and it may be sufficient if the input is only given as an initial state, and the system is allowed to evolve according to its internal dynamical rule, and this will characterize the normal model of the dynamical system
E transition rules for the dynamical system are usually determined by a set of parameters
Summary
Introduction and DefinitionsA dynamical system might be an arrangement that grows in time via the iterative application of a dynamical rule. e evolution rule shows the transformation of the real state under its conditions as well as possibly earlier conditions.e fact that state evolutions must be based on the system’s states suggests that the dynamics are recursive. M} be a chain of topological dynamics in a dynamical manifolds; we define the limit topological dynamics as limm⟶∞(ηm(Nm−1, tm−1)) limm⟶∞ηm If the limit topological dynamics reduces the dimensions of a dynamical manifold, we call it the limit conditional topological dynamical and is denoted by limm⟶∞(ξm(Nn, tnm)) (Nn− 1, tn− 1) where (Nn, tnm) is an n-dimensional dynamical manifold.
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