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This article presents a formula for calculating the volume entropy of a family of metrics on Euclidean space that generalize the metrics of the SOL group and the logarithmic model of Hyperbolic space.

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arxiv.org

Garcia-Lara, R. I. (2024). Volume Entropy of a Family of Rank One, Split-Solvable Lie Groups of Abelian Type. arXiv:2410.10631v1 [math.DG].

This paper aims to derive a formula for the volume entropy of a family of metrics on Euclidean space that generalize the left-invariant metric of the SOL group and the metric of the logarithmic model of Hyperbolic space. The research also aims to address a conjecture related to the volume entropy of a family of 3-manifolds that interpolate between the SOL group and hyperbolic space.

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by Rene Garcia-... at **arxiv.org** 10-15-2024

Deeper Inquiries

Volume entropy, while fundamentally a concept in Riemannian geometry, finds profound applications in areas beyond, such as theoretical physics and dynamical systems. This is because it captures the rate at which the volume of a space grows as we explore it further and further from a given point. This notion of "growth at infinity" has deep connections with the behavior of physical systems and the evolution of dynamical systems over time. Here's a glimpse into how volume entropy plays a role:
Theoretical Physics:
Cosmology and General Relativity: In cosmology, volume entropy is used to study the large-scale structure of the universe and the evolution of cosmological models. For instance, the entropy of the cosmological horizon is related to the number of degrees of freedom within the observable universe.
Black Hole Thermodynamics: The entropy of a black hole, as proposed by Bekenstein and Hawking, is proportional to the area of its event horizon. This entropy is thought to be related to the number of quantum states available to the black hole, and volume entropy plays a role in understanding the microscopic origin of this entropy.
Quantum Field Theory: In quantum field theory, particularly in curved spacetime, volume entropy is related to the entanglement entropy of quantum fields. This connection has implications for understanding the nature of quantum entanglement and the role of gravity in quantum theory.
Dynamical Systems:
Topological Entropy: Volume entropy is closely related to the concept of topological entropy in dynamical systems. Topological entropy measures the exponential growth rate of the number of distinguishable orbits of a dynamical system. In many cases, volume entropy provides an upper bound for topological entropy.
Ergodic Theory: In ergodic theory, volume entropy is used to study the statistical properties of dynamical systems. For example, it can be used to characterize the mixing properties of flows on Riemannian manifolds.
Geodesic Flows: The volume entropy of a Riemannian manifold is intimately connected to the dynamics of the geodesic flow on the unit tangent bundle of the manifold. Understanding the growth of volumes helps in characterizing the chaotic behavior of geodesics.
In essence, volume entropy acts as a bridge between the geometry of a space and the dynamics that unfold within it. Its applications extend to diverse areas of physics and mathematics, providing insights into the fundamental nature of space, time, and the evolution of systems.

Yes, while volume entropy is a powerful invariant, alternative geometric invariants might capture the growth of geodesic balls more effectively for specific families of metrics. Here are a few examples:
Diameter Growth: Instead of volume, one could consider the growth rate of the diameter of geodesic balls. This invariant, known as diameter growth, can be more sensitive to the finer geometric features of a space, especially in cases where the volume growth is slow.
Filling Volume: The filling volume of a loop in a Riemannian manifold is the minimal volume of a disk bounding the loop. The asymptotic behavior of filling volumes for loops of increasing lengths provides information about the geometry at large scales.
Isoperimetric Constants: Isoperimetric constants relate the volume of a region to the volume of its boundary. Different isoperimetric constants, such as the Cheeger constant, can provide finer information about the growth of geodesic balls and the overall shape of the space.
Eigenvalue Asymptotics: The asymptotic distribution of the eigenvalues of the Laplace-Beltrami operator on a Riemannian manifold is closely related to the volume growth. In some cases, the eigenvalues can provide more refined information about the geometry than the volume entropy alone.
The choice of the most effective invariant depends on the specific family of metrics and the geometric properties one wants to emphasize. For instance, diameter growth might be more suitable for studying spaces with a lot of "narrow necks," while filling volume growth is relevant for understanding the "filling properties" of the space.

The connection between information-theoretic entropy and volume entropy hinges on the idea of "uncertainty" or "complexity" associated with exploring a space.
Information Theory: In information theory, entropy quantifies the uncertainty about the outcome of a random variable. A high entropy implies greater uncertainty in predicting the outcome.
Volume Entropy: Volume entropy, on the other hand, measures how rapidly the volume of geodesic balls grows as their radius increases. A high volume entropy suggests that the space becomes increasingly "spread out" and "complex" as we venture further from a given point.
Here's how the notion of "uncertainty" bridges the two:
Exploring a Space: Imagine exploring a space by walking along geodesic paths. In a space with low volume entropy, the volume we encounter grows slowly, implying that the space is relatively "predictable" and "homogeneous." Conversely, in a space with high volume entropy, the volume grows rapidly, suggesting a more "uncertain" and "complex" structure, with potentially many different directions and features to encounter.
Distinguishability of Points: A high volume entropy implies that there are many geodesic balls of a given radius that are "far apart" from each other. This means that many points in the space are "distinguishable" from each other at that scale, leading to a greater degree of "uncertainty" about the global structure of the space.
Information Content of Geodesics: Consider the information required to describe a geodesic path in the space. In a space with high volume entropy, we would need more information to specify a geodesic path accurately, as there are more potential directions and branches it could take.
In essence, volume entropy can be viewed as a measure of the "information content" or "complexity" of a space as seen from the perspective of an explorer traversing geodesic paths. A higher volume entropy reflects a greater degree of "uncertainty" about the global structure and a richer set of possibilities for exploration.

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