Constructions of the invariant sets and invariant measures

Abstract

In this paper, we will discuss the construction problems about the invariant sets and invariant measures of continuous maps which map complexes into themselves, using simplicial approximation and Markov chains. In [7], the author defined a matrix by using r-normal subdivision of the n-dimensional unit cube, considered it a Markov matrix, and constructed the invariant set and invariant measure. In fact, the matrix he defined is not Markov matrix generally. So we will give [7] and amendment in the last part of this paper. We also construct an invariant set that is the chain-recurrent set of the map by means of a non-negative matrix which only depends on the map. At last, we will prove the higher dimension Banach variation formula that can simplify the transition matrix.