On the equivalence between noncollapsing and bounded entropy for ancient solutions to the Ricci flow

As a continuation of a previous paper, we prove Perelman’s assertion, that is, for ancient solutions to the Ricci flow with bounded nonnegative curvature operator, uniformly bounded entropy is equivalent to κ-noncollapsing on all scales. We also establish an equality between the asymptotic entropy and the asymptotic reduced volume, which is a result similar to a paper by Xu (2017), where he assumes the Type I curvature bound.

Asymptotic entropy of transformed random walks

We consider general transformations of random walks on groups determined by Markov stopping times and prove that the asymptotic entropy (respectively, rate of escape) of the transformed random walks is equal to the asymptotic entropy (respectively, rate of escape) of the original random walk multiplied by the expectation of the corresponding stopping time. This is an analogue of the well-known Abramov formula from ergodic theory; its particular cases were established earlier by Kaimanovich [Differential entropy of the boundary of a random walk on a group. Uspekhi Mat. Nauk38(5(233)) (1983), 187–188] and Hartman et al [An Abramov formula for stationary spaces of discrete groups. Ergod. Th. & Dynam. Sys.34(3) (2014), 837–853].

ASYMPTOTIC ENTANGLEMENT IN THE PARAMETRIZED HADAMARD WALK

We study asymptotic entanglement properties of the Hadamard walk with phase parameters on the line using the Fourier representation. We use the von Neumann entropy of the reduced density operator to quantify entanglement between the coin and position degrees of freedom. We investigate obtaining exact expressions for the asymptotic entropy of entanglement, for different classes of initial conditions. We also determine under which conditions the asymptotic entropy of entanglement can be characterized as full, intermediate, or minimum.