We show vanishing results about the infimum of the topological entropy of the geodesic flow of homogeneous smooth four manifolds. We prove that any closed oriented geometric four manifold has zero minimal entropy if and only if it has zero simplicial volume. We also show that if a four manifold M admits a geometric decomposition, in the sense of Thurston, and does not have geometric pieces mode...

We present a method to compute rigorous upper bounds for the topological entropy h(T ,A) of a continuous map T with respect to a fixed (coarse) partition of the phase space A. Long trajectories are not used; rather a single application of T to the phase space produces a topological Markov chain which contains all orbits of T , plus some additional spurious orbits. By considering the Markov chai...

We consider partially hyperbolic diffeomorphisms on compact manifolds where the unstable and stable foliations stably carry some unique nontrivial homologies. We prove the following two results: if the center foliation is one dimensional, then the topological entropy is locally a constant; and if the center foliation is two dimensional, then the topological entropy is continuous on the set of a...

For general asymptotically sub-additive potentials (resp. asymptotically additive potentials) on general topological dynamical systems, we establish some variational relations between the topological entropy of the level sets of Lyapunov exponents, measuretheoretic entropies and topological pressures in this general situation. Most of our results are obtained without the assumption of the exist...

For general asymptotically sub-additive potentials (resp. asymptotically additive potentials) on general topological dynamical systems, we establish some variational relations between the topological entropy of the level sets of Lyapunov exponents, measuretheoretic entropies and topological pressures in this general situation. Most of our results are obtained without the assumption of the exist...

We study topological properties of quasi-particle states in the non-Abelian quantum Hall states. We apply a skein-theoretic method to the Read–Rezayi state whose effective theory is the SU(2)K Chern–Simons theory. As a generalization of the Pfaffian (K = 2) and the Fibonacci (K = 3) anyon states, we compute the braiding matrices of quasi-particle states with arbitrary spins. Furthermore we prop...

Given a saddle fixed point of a surface diffeomorphism, its stable and unstable curves W S and W U often form a homoclinic tangle. Given such a tangle, we use topological methods to find periodic points of the diffeomorphism, using only a subset of the tangle with finitely many points of intersection, which we call a trellis. We typically obtain exponential growth of periodic orbits, symbolic d...

We provide a rigorous lower bound for the topological entropy of planar diffeomorphisms in terms of the geometry of finite pieces of stable and unstable manifolds of hyperbolic periodic points. Our results suggest the possibility of writing computer programs which would automate the estimation of reasonable approximations for the topological entropy of mappings and differential equations. Apply...

We realize the logarithm of the third smallest known Salem number as the topological entropy of a K3 surface automorphism with a Siegel disk and a pointwisely fixed curve at the same time. We also show the logarithm of the Lehmer number, the smallest known Salem number, is not realizable as the topological entropy of any Enriques surface automorphism. These results are entirely inspired by McMu...