Heterodimensional cycles, partial hyperbolicity and limit dynamics

Partial Hyperbolicity and Specification

We study the specification property for partially hyperbolic dynamical systems. In particular, we show that if a partially hyperbolic diffeomorphism has two saddles with different indices, and the stable manifold of one of these saddles coincides with the strongly stable leaf, then it does not satisfy the specification property.

Nongeneric Bifurcations near Heterodimensional Cycles with Inclination Flip in R

Nongeneric bifurcation analysis near rough heterodimensional cycles associated to two saddles in R4 is presented under inclination flip. By setting up local moving frame systems in some tubular neighborhood of unperturbed heterodimensional cycles, we construct a Poincaré return map under the nongeneric conditions and further obtain the bifurcation equations. Coexistence of a heterodimensional c...

Hopf Bifurcations and Limit Cycles in Evolutionary Network Dynamics

The replicator-mutator equations from evolutionary dynamics serve as a model for the evolution of language, behavioral dynamics in social networks, and decision-making dynamics in networked multi-agent systems. Analysis of the stable equilibria of these dynamics has been a focus in the literature, where symmetry in fitness functions is typically assumed. We explore asymmetry in fitness and show...

Entropic Stability beyond Partial Hyperbolicity

We analyze a class of deformations of Anosov diffeomorphisms: these C-small, but C-macroscopic deformations break the topological conjugacy class but leave the high entropy dynamics unchanged. More precisely, there is a partial conjugacy between the deformation and the original Anosov system that identifies all invariant probability measures with entropy close to the maximum. We also establish ...

Valuations and Hyperbolicity in Dynamics