# On Non-Linear Markov Operators: surjectivity vs orthogonal preserving property

## TL;DR

This paper investigates nonlinear Markov operators, specifically polynomial stochastic operators, establishing that their surjectivity is equivalent to possessing an orthogonal preserving property.

## Contribution

It introduces the concept of orthogonal preserving polynomial stochastic operators and proves their surjectivity is equivalent to this property.

## Key findings

- Surjectivity of nonlinear Markov operators is equivalent to orthogonal preserving property.
- Introduces orthogonal preserving polynomial stochastic operators.
- Provides theoretical foundation linking surjectivity and orthogonal preservation.

## Abstract

In the present paper, we consider nonlinear Markov operators, namely polynomial stochastic operators. We introduce a notion of orthogonal preserving polynomial stochastic operators. The purpose of this study is to show that surjectivity of nonlinear Markov operators is equivalent to their orthogonal preserving property.