Constructions of the invariant sets and invariant measures

Abstract

In this paper, we will di~uss 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.