Introduction
The relation between robust transitivity and robust ergodicity for conservative diffeomorphisms forms the crux of this paper. It is simple to verify that if m:= Lebesgue measure is ergodic or f preserves any ergodic probability measure which imparts a positive mass to open balls, then f is topologically transitive. However, the converse implication does not hold goround, that is, transitivity does not guarantee ergodicity. Through this paper, our main goal is to relate these two notions when they persist in the neighbourhood of a diffeomorphism.

Continuity is a necessary requirement for proving ergodicity and even for C₁ Anosov diffeomorphisms preserving volume, the ergodicity is not verified. This paper extends to define C₁-robustly transitive (resp. C₁-robustly ergodic), which is an essential entity in the open set in C₁ topology U⊂Diff₁(M)(resp.U⊂Diff₁ m(M)) such that any g in U(resp. in U∩Diff₁₊(M))is also topologically transitive (resp. ergodic).

Case Analysis
The paper further leverages case studies to extrapolate the implications of robustly transitive diffeomorphisms. For example, for surface diffeomorphisms robustly transitive diffeomorphisms prove to be Anosov. However, in the case of three-manifolds, the diffeomorphisms may be non-Anosov. This leads to various tactical splits in the tangent bundle of the ambient manifold.

Questions
The paper further poses some interesting questions like - Can a C₁-robustly transitive be approximated by robustly ergodic diffeomorphisms? How feasible is it to approximate any robustly transitive diffeomorphism by another whose Lyapunov exponents are nonzero in a full Lebesgue measure set?

Theorems and Definitions
Throughout the paper, we introduce various theorems and definitions that dive deep into understanding ergodic diffeomorphisms, robust transitivity diffeomorphisms and more. This paper will prove helpful in relation to mathematical nuances in higher dimensions, addressing questions around how ergodic diffeomorphisms form a generic subset of U∩Diff₁₊ m, among others.

Conclusion
This paper offers a comprehensive exploration of the correlation between robust transitivity and robust ergodicity for conservative diffeomorphisms across various dimensions. It moves from establishing clear definitions to formulating theorems, all aiming to enhance understanding around the two intriguing concepts and their intersection. This research is far-reaching, and it would be intriguing to see how further work in this field evolves.