# On the rotation sets of generic homeomorphisms on the torus $\mathbb T^d$

## TL;DR

This paper investigates the properties of rotation sets for generic homeomorphisms on tori, revealing generic convexity, complex pointwise behaviors, and measure-theoretic richness in both conservative and dissipative settings.

## Contribution

It establishes the generic convexity of rotation sets for $C^0$-generic conservative homeomorphisms on tori and describes the prevalence of wild pointwise rotation sets with full pressure and dimension.

## Key findings

- Existence of a residual set with wild rotation behavior in conservative homeomorphisms.
- Convexity of rotation sets for generic conservative homeomorphisms on $ ext{T}^d$.
- Full topological pressure and metric mean dimension for wild rotation sets.

## Abstract

We study the rotation sets for homeomorphisms homotopic to the identity on the torus $\mathbb T^d$, $d\ge 2$. In the conservative setting, we prove that there exists a Baire residual subset of the set $\text{Homeo}_{0, \lambda}(\mathbb T^2)$ of conservative homeomorphisms homotopic to the identity so that the set of points with wild pointwise rotation set is a Baire residual subset in $\mathbb T^2$, and that it carries full topological pressure and full metric mean dimension. Moreover, we prove that for every $d\ge 2$ the rotation set of $C^0$-generic conservative homeomorphisms on $\mathbb T^d$ is convex. Related results are obtained in the case of dissipative homeomorphisms on tori. The previous results rely on the description of the topological complexity of the set of points with wild historic behavior and on the denseness of periodic measures for continuous maps with the gluing orbit property.