Symbolic dynamics for non-uniformly hyperbolic flows in high dimension

We construct symbolic dynamics for flows with positive speed in any dimension: for each $\chi>0$, we code a set that has full measure for every invariant probability measure which is $\chi$--hyperbolic. In particular, the coded set contains all hyperbolic periodic orbits with Lyapunov exponents outside of $[-\chi,\chi]$. This extends the recent work of Buzzi, Crovisier, and Lima for three dimensional flows with positive speed. As an application, we code homoclinic classes of measures by suspensions of irreducible countable Markov shifts, and prove that each such class has at most one probability measure that maximizes the entropy.