Besides magnetic and charge order, regular arrangements of orbital occupation constitute a fundamental order parameter of condensed matter physics. Even though orbital order is difficult to identify directly in experiments, its presence was firmly es
tablished in a number of strongly correlated, three-dimensional Mott insulators. Here, reporting resonant X-ray scattering experiments on the layered Van der Waals compound $1T$-TiSe$_2$, we establish the emergence of orbital order in a weakly correlated, quasi-two-dimensional material. Our experimental scattering results are consistent with first-principles calculations that bring to the fore a generic mechanism of close interplay between charge redistribution, lattice displacements, and orbital order. It demonstrates the essential role that orbital degrees of freedom play in TiSe$_2$, and their importance throughout the family of correlated Van der Waals materials.
In this paper we introduce the definition of topological $r$-pressure of free semigroup actions on compact metric space and provide some properties of it. Through skew-product transformation into a medium, we can obtain the following two main results
. 1. We extend the result that the topological pressure is the limit of topological $r$-pressure incite{C} to free semigroup actions ($rto 0$). 2. Let $f_i,$ $i=0, 1, cdots, m-1$, be homeomorphisms on a compact metric space. For any continuous function, we verify that the topological pressure of $f_0, cdots, f_{m-1}$ equals the topological pressure of $f_0^{-1}, cdots, f_{m-1}^{-1}.$
