arXiv subjects
Publications and source records attributed to Thomas Aloysius O'Hare.
We show that for closed negatively curved Riemannian manifolds whose length spectrum is exponentially separated, the length spectrum is a local isometry invariant. In particular, this holds for a dense set of negatively curved metrics. The main tool used in the proof is a finitary version of the Livshits theorem with an exponential tail.
We prove effective equidistribution theorems for (weighted) packets of closed periodic orbits for Anosov flows. In particular, for the case of contact Anosov flows on three-dimensional manifolds, we show that the Bowen packets equidistribute at an exponential rate.
Let $f,g$ be $C^2$ area-preserving Anosov diffeomorphisms on $\mathbb{T}^2$ which are topologically conjugate by a homeomorphism $h$ ($hf=gh$). We assume that the Jacobian periodic data of $f$ and $g$ are matched by $h$ for all points of some large period $N\in\mathbb{N}$. We show that $f$ and $g$ are ``approximately smoothly conjugate." That is, there exists a $C^{1+α}$ diffeomorphism $\overline{h}_N$ such that $h$ and $\overline{h}_N$ are $C^0$ exponentially close in $N$, and $f$ and $f_N:=\overline{h}_N^{-1}g\overline{h}_N$ are $C^1$ exponentially close in $N$. Moreover, the rates of convergence are uniform among different $f,g$ in a $C^2$ bounded set of Anosov diffeomorphisms. The main idea in constructing $\overline{h}_N$ is to do a ``weighted holonomy" construction, and the main technical tool in obtaining our estimates is a uniform effective version of Bowen's equidistribution theorem of weighted discrete orbits to the SRB measure.