# Sets of invariant measures and Cesaro stability

## TL;DR

This paper investigates the properties of invariant measure sets for continuous maps on compact metric spaces, showing that typical maps are statistically structurally stable using Kantorovich and Hausdorff metrics.

## Contribution

It introduces a framework for analyzing invariant measure sets with specific metrics and proves that typical maps exhibit statistical stability in this context.

## Key findings

- Typical maps are continuity points of the invariant measure set function.
- For typical maps, points are statistically structurally stable.
- The approach combines Kantorovich and Hausdorff metrics for measure and set analysis.

## Abstract

Sets of invariant measures are considered for continuous maps of a metric compact set. We take Kantorovich metric to calculate distance between measures and Hausdorff metrics to calculate distance between compact sets. Consider the function that makes correspondence between a continuous map and the set of all its Borel probability invariant measures. We demonstrate that a typical map is a continuity point of that function. Using approaches of Takens' tolerance stability theory we provide some corollaries that demonstrate that for a typical map points are structurally stable in a statistical sense.