In this article, the Hausdorff dimension is estimated using the box-counting dimension and the information dimension. It is shown that the former is an upper bound for the Hausdorff dimension, while the latter is a lower bound for the box-counting dimension, and that the information dimension is an upper bound for the Hausdorff dimension. Additionally, the dimension of the Henon attractor is estimated using the box-counting dimension.

We revise the dynamics of interacting vector-like dark energy, a theoretical framework proposed to explain the accelerated expansion of the universe. By investigating the interaction between vector-like dark energy and dark matter, we analyze its effects on the cosmic expansion history and the thermodynamics of the accelerating universe. Our results demonstrate that the presence of interaction significantly influences the evolution of vector-like dark energy, leading to distinct features in its equation of state and energy density. We compare our findings with observational data and highlight the importance of considering interactions in future cosmological studies.

We investigate the cosmological dynamics of a homogeneous scalar field non-minimally coupled to torsion gravity, which also interacts with cold dark matter through energy and momentum transfer. The matter and radiation perfect fluids are modeled using the Sorkin-Schutz formalism. We identify scaling regimes of the field during both the radiation and matter eras. Additionally, we discovered a field-dominated scaling attractor; however, it does not exhibit accelerated expansion, making it unsuitable for describing dark energy. Nevertheless, we find two attractor solutions that do exhibit accelerated expansion: one is a quintessence-like fixed point, and the other is a de Sitter fixed point.