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Abstract We study the degree of an L-Lipschitz map between Riemannian manifolds, proving new upper bounds and constructing new examples. For instance, if $$X_k$$ is the connected sum of k copies of $$\mathbb CP^2$$for$$k \ge 4$$, then we prove that the maximum degree of an L-Lipschitz self-map of $$X_k$$ is between $$C_1 L^4 (\log L)^{-4}$$ and $$C_2 L^4 (\log L)^{-1/2}$$. More generally, we divide simply connected manifolds into three topological types with three different behaviors. Each type is defined by purely topological criteria. For scalable simply connected n-manifolds, the maximal degree is $$\sim L^n$$. For formal but nonscalable simply connectedn-manifolds, the maximal degree grows roughly like $$L^n (\log L)^{-\theta (1)}$$. And for nonformal simply connected n-manifolds, the maximal degree is bounded by $$L^\alpha $$ for some $$\alpha < n$$.
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This content will become publicly available on July 1, 2026
New classes of quasigeodesic Anosov flows in $3$-manifolds
Abstract Quasigeodesic behavior of flow lines is a very useful property in the study of Anosov flows. Not every Anosov flow in dimension three is quasigeodesic. In fact, until recently, up to orbit equivalence, the only previously known examples of quasigeodesic Anosov flows were suspension flows. In a recent article, the second author proved that an Anosov flow on a hyperbolic 3-manifold is quasigeodesic if and only if it is non-$$\mathbb {R}$$-covered, and this result completes the classification of quasigeodesic Anosov flows on hyperbolic 3-manifolds. In this article, we prove that a new class of examples of Anosov flows are quasigeodesic. These are the first examples of quasigeodesic Anosov flows on 3-manifolds that are neither Seifert, nor solvable, nor hyperbolic. In general, it is very hard to show that a given flow is quasigeodesic and, in this article, we provide a new method to prove that an Anosov flow is quasigeodesic.
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- Award ID(s):
- 2054909
- PAR ID:
- 10612130
- Publisher / Repository:
- Cambridge University Press
- Date Published:
- Journal Name:
- Ergodic Theory and Dynamical Systems
- Volume:
- 45
- Issue:
- 7
- ISSN:
- 0143-3857
- Page Range / eLocation ID:
- 2095 to 2131
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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