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Let T be an ergodic endomorphism of the probability space X and let f:X->R be a real-valued measurable function. Then for almost every x in X, we have 1/nsum_(j=1)^nf ...

Any symmetric polynomial (respectively, symmetric rational function) can be expressed as a polynomial (respectively, rational function) in the elementary symmetric ...

A special case of a flag manifold. A Grassmann manifold is a certain collection of vector subspaces of a vector space. In particular, g_(n,k) is the Grassmann manifold of ...

A homogeneous space M is a space with a transitive group action by a Lie group. Because a transitive group action implies that there is only one group orbit, M is isomorphic ...

The Lyapunov characteristic exponent [LCE] gives the rate of exponential divergence from perturbed initial conditions. To examine the behavior of an orbit around a point ...

Every smooth nonzero vector field on the 3-sphere has at least one closed orbit. The conjecture was proposed in 1950 and proved true for Hopf maps. The conjecture was ...

Transitivity is a result of the symmetry in the group. A group G is called transitive if its group action (understood to be a subgroup of a permutation group on a set Omega) ...

A group action G×X->X is transitive if it possesses only a single group orbit, i.e., for every pair of elements x and y, there is a group element g such that gx=y. In this ...

Let x_0 be a rational number in the closed interval [0,1], and generate a sequence using the map x_(n+1)=2x_n (mod 1). (1) Then the number of periodic map orbits of period p ...

A spiral that gives the solution to the central orbit problem under a radial force law r^..=-mu|r|^(-3)r^^, (1) where mu is a positive constant. There are three solution ...