Related papers: Minimal volume entropy and fiber growth
In this note we give examples of Hamiltonian diffeomorphisms which are on one hand dynamically complicated, for instance with positive topological entropy, and on the other hand minimal from the perspective of Floer theory. The minimality…
We define a Bowen-Series like map for every geometric presentation of a co-compact surface group and we prove that the volume entropy of the presentation is the topological entropy of this particular (circle) map. Finally we find the…
In the theory of surface diffeomorphisms relative to homoclinic and heteroclinic orbits, it is possible to compute a one-dimensional representative map for any irreducible isotopy class. The topological entropy of this graph representative…
We consider an entropy-type invariant which measures the polynomial volume growth of submanifolds under the iterates of a map, and we establish sharp uniform lower bounds of this invariant for the following classes of symplectomorphisms of…
Inspired by the idea of Colding-Minicozzi in [CM1], we define (mean curvature flow) entropy for submanifolds in a general ambient Riemannian manifold. In particular, this entropy is equivalent to area growth of a closed submanifold in a…
We introduce the notion of topological entropy of a formal languages as the topological entropy of the minimal topological automaton accepting it. Using a characterization of this notion in terms of approximations of the Myhill-Nerode…
We prove that the number of 3-dimensional simplicial complexes having the spherical topology grows exponentially as a function of a volume. It is suggested that the 3d simplicial quantum gravity has qualitatively the same phase structure as…
Quantum gravity is expected to impose constraints on the moduli spaces of massless fields that can arise in effective quantum field theories. A recent proposal asserts that the asymptotic volume growth of these spaces is severely…
This paper studies whether the presence of a perimeter minimizing set in a Riemannian manifold $(M,g)$ forces an isometric splitting. We show that this is the case when $M$ has non-negative sectional curvature and quadratic volume growth at…
We establish conditions for a continuous map of nonzero degree between a smooth closed manifold and a negatively curved manifold of dimension greater than four to be homotopic to a smooth cover, and in particular a diffeomorphism when the…
We investigate compactness phenomena involving free boundary minimal hypersurfaces in Riemannian manifolds of dimension less than eight. We provide natural geometric conditions that ensure strong one-sheeted graphical subsequential…
A well-known question by Gromov asks whether the vanishing of the simplicial volume of oriented closed connected aspherical manifolds implies the vanishing of the Euler characteristic. We study various versions of Gromov's question and…
This paper studies minimal surface entropy (the exponential asymptotic growth of the number of minimal surfaces up to a given value of area) for negatively curved metrics on hyperbolic $3$-manifolds of finite volume, particularly its…
A finite volume symplectic manifold is said to have "packing stability" if the only obstruction to symplectically embedding sufficiently small balls is the volume obstruction. Packing stability has been shown in a variety of cases and it…
For Hermitian connections on a Hermitian complex line bundle over a Riemannian manifold $(X,g)$, we can define the ``volume", which can be considered to be the ``mirror" of the standard volume for submanifolds. We call the critical points…
Let $f$ be a $C^2$ diffeomorphism on a compact manifold. Ledrappier and Young introduced entropies along unstable foliations for an ergodic measure $\mu$. We relate those entropies to covering numbers in order to give a new upper bound on…
It is proved that the fundamental group of a complete Riemannian manifold with nonnegative Ricci curvature and certain volume growth conditions is trivial or finite.
We prove minimal entropy rigidity for complete, finite volume manifolds locally isometric to a product of rank one symmetric spaces of dimension at least 3: the locally symmetric metric uniquely minimizes (normalized) entropy among all…
We prove the existence of manifolds with almost maximal volume entropy which are not hyperbolic.
Topological entropy is not lower semi-continous: small perturbation of the dynamical system can lead to a collapse of entropy. In this note we show that for some special classes of dynamical systems (geodesic flows, Reeb flows, positive…