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Approximations of the Long-Time Dynamics of the Navier-Stokes Equations
Published in K. D. Elworthy, W. Norrie Evenitt, E. Bruce Lee, Differential equations, dynamical systems, and control science, 2017
Victor A. Pliss, George R. Sell
The principal dynamical objects of interest in this paper are compact, invariant sets, and especially, compact attractors, with suitable hyperbolic structures. In Section 5 we shall give precise definitions of a weakly hyperbolic and a weakly, normally hyperbolic invariant set. As explained below, these concepts generalize and include the notion of an Anosov flow, see Markus (1961, 1971), Smale (1967), and Arnold (1983), as well as a normally hyperbolic manifold, see Sacker (1969), Fenichel (1971), Hirsch, Pugh and Shub (1977), and Pliss (1977).
A modified Riemannian Halpern algorithm for nonexpansive mappings on Hadamard manifolds
Published in Optimization, 2022
Teng-Teng Yao, Ying-Hui Li, Yong-Shuai Zhang, Zhi Zhao
We consider the examples given in [16]. Let be the hyperbolic manifold defined by (7). Given a point and a unit tangent vector , the normalized geodesic starting from is given by The distance on is given by Equation (60) implies that the exponential map is given by for t>0, , and any unit tangent vector . The inverse of the exponential map is given by for . For parallel translation on , we use the following approximation [41, p.174]: for and .
Phase transitions for the geodesic flow of a rank one surface with nonpositive curvature
Published in Dynamical Systems, 2021
K. Burns, J. Buzzi, T. Fisher, N. Sawyer
We note that little is known about phase transitions for the geodesic flow of compact rank one manifolds with nonpositive sectional curvature in higher dimension. The Heintze example [3, Example 4] shows that at least two different behaviours are possible. Heintze's example is constructed by taking two copies of a compact hyperbolic manifold of dimension with finite volume and one cusp, cutting off the ends of the cusps, and gluing the two halves together. The resulting manifold has nonpositive curvature and contains a flat totally geodesically isometrically embedded torus T of dimension n−1. Moreover, it can be arranged that each plane spanned by a vector orthogonal to T and a vector tangent to T has sectional curvature zero or that all such planes have a negative curvature. In the first case, we see the same phase transition at q = 1 as in the surface case. In the second case, as explained in [6, Section 10.2], it can be arranged that has a unique equilibrium state for each and there are no phase transitions. On the other hand, Theorem D of [6] proves that the family does not have a phase transition at q = 0 for the geodesic flow of any compact rank 1 manifold with nonpositive sectional curvature.