# Inverse shadowing and related measures

## TL;DR

This paper introduces and analyzes weaker forms of inverse shadowing in discrete dynamical systems, linking ergodic properties to stability and measure continuity, and broadening understanding of system robustness.

## Contribution

It defines the Ergodic Inverse Shadowing property, shows its implications for measure continuity, and relates the Individual Inverse Shadowing property to structural stability.

## Key findings

- Ergodic Inverse Shadowing implies continuity of invariant measures.
- Systems with hyperbolic nonwandering sets have Ergodic Inverse Shadowing.
- Individual Inverse Shadowing relates to structural and $\Omega$-stability.

## Abstract

We study various weaker forms of inverse shadowing property for discrete dynamical systems on a smooth compact manifold. First, we introduce the so-called Ergodic Inverse Shadowing property (Birhhoff averages of continuous functions along the exact trajectory and the approximating one are close). We demonstrate that this property implies continuity of the set of invariant measures in Hausdorff metrics. We show that the class of systems with Ergodic Inverse Shadowing is quite broad, it includes all diffeomorphisms with hyperbolic nonwandering sets. Secondly, we study the so-called Individual Inverse Shadowing (any exact trajectory can be traced by approximate ones but this shadowing is not uniform with respect to selection of the initial point of the trajectory). We demonstrate that this property is closely related to Structural Stability and $\Omega$-stability of diffeomorphisms.