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The grand potential isFor a homogeneous extensive system, $\Phi =-pV$.

The grand canonical partition function is

An interval map $F:I\to I$ has a horseshoe when there are an open interval $J\subseteq I$ and disjoint open intervals $K_0,K_1\subseteq J$ such thatIterating inverse branches produces full two-symbol itinerary dynamics.

Ice sheet dynamics describes the gravity-driven deformation and basal motion of large terrestrial ice masses together with their mass balance.

Maximizing the Gibbs entropy under normalization alone gives a uniform distribution. Adding a fixed mean-energy constraint gives the Boltzmann distribution $p_n=Z^{-1}{e}^{-\beta E_n}$.

An equatorial wave is trapped near the equator by the sign change of the Coriolis parameter. Its meridional structures are Gaussian-weighted Hermite polynomials.

A Rossby wave is a low-frequency wave restored by the spatial variation of planetary or background potential vorticity. On a beta plane its phase commonly propagates westward.

Ocean circulation comprises wind-driven, buoyancy-driven, and tidal motions of the ocean over a wide range of spatial and temporal scales.

Geostrophic adjustment is the radiation of inertia-gravity waves from an unbalanced disturbance, leaving a slower flow in geostrophic balance while conserving linearized potential vorticity.

The quasi-geostrophic approximation describes slowly evolving, nearly geostrophic flow with small Rossby number. Its dynamics are governed by advection and forcing of potential vorticity.

A rotating fluid is described in a rotating reference frame, where the Coriolis acceleration and centrifugal acceleration supplement the physical forces.

The Boussinesq approximation treats density as constant except where a small density variation multiplies gravity to produce buoyancy. The velocity remains incompressible.

An action is properly discontinuous when every compact set meets only finitely many of its translates. For an isometric action on a proper metric space, this gives the finiteness needed to extract a constant group element from a sequence with bounded source and target.

For a discrete group action on a locally compact space, a standard proper-discontinuity condition is that only finitely many group elements translate a given compact set to meet itself.

A group action on a space is cocompact when its quotient is compact, equivalently when some compact set has translates covering the space.

A metric space is proper when every closed bounded subset is compact.

For an isometry $g$ of a metric space, its displacement function is $d_g(x)=d(x,gx)$. It is $2$-Lipschitz.

A subset $Y$ of a geodesic metric space is $K$-quasiconvex when every metric geodesic between points of $Y$ lies in the closed $K$-neighborhood of $Y$.

A map $f:X\to Y$ is a $(\lambda ,\epsilon )$-quasi-isometric embedding whenIt is a quasi-isometry when every point of $Y$ lies a uniformly bounded distance from its image.

A Fuchsian group is a discrete subgroup of ${\mathrm{PSL}}_2(\mathbb{R})$, acting by orientation-preserving isometries of the hyperbolic plane. It is non-elementary when it has no finite orbit in the hyperbolic plane or its ideal boundary.

A group action on a tree is a homomorphism to the tree's combinatorial isometry group. Barycentric subdivision removes edge inversions without changing the essential action.