## Self orbit equivalences for Anosov flows in 3d

October 10, 2018

Our note with Thomas Barthelmé was accepted to Math Research Letters yesterday. (Hat tip to an anonymous referee for providing very helpful and thorough reports.) This note was motivated by my work with Tom Farrell on topological rigidity of Anosov bundles, which are fiber bundles equipped with fiberwise Anosov flows.

Example Let ${S}$ be a surface of genus ${\ge2}$ and let ${S\rightarrow E\rightarrow X}$ be a fiber bundle. Because the space of negatively curved metrics on ${S}$ is contractible the fibers of the bundle can be equipped with continuously varying metrics of negative curvature. Then such fiberwise metric yields a fiberwise Anosov geodesic flow on the associated bundle

$\displaystyle T^1S\rightarrow T^1E\rightarrow X,$

whose fiber is the unit tangent bundle of ${S}$.

## Penn State workshop 2018

October 9, 2018

Below the fold is a quick photo report from the conference. According to the organizers there were about 150 people in attendance.

## On Beyond Hyperbolicity

October 1, 2018

Today Ruth Charney delivered her first Rado Lecture here at OSU. The lecture was done beautifully. To my surprise, I learned that smooth dynamicists are not the only people who love the word “beyond.”

## PennState Workshop 2018

September 9, 2018

The PennSate workshop of 2018 edition is coming up soon and it will be dedicated to the memory of Anatole Katok. The line up of speakers is stellar and my feeling is that the attendance would be record breaking. Another memorial event is being planned for August 2019 in Bedlewo Conference Center — 2020 Vision for Dynamics (there is not much on the page yet).

Also, this year PennState math department will begin a worldwide search for newly endowed Katok’s Chair at PennState. Michael Brin, Sergey Ferleger and Aleksey Kononenko have made a gift to endow this new position at PennState Math Department as well as to provide permanent support to programs run by the Center of Dynamics in Geometry. Some more information can be found in PennState press-release.

## Rigidity of partially hyperbolic automorphisms from volume Lyapunov spectra

August 27, 2018

Last week Boris Kalinin, Victoria Sadovskaya and myself have posted a short preprint on Lyapunov spectrum stability on the arxiv. We have been inspired by work Saghin-Yang discussed here earlier. Namely we have generalized the local rigidity Theorem for Anosov automorphisms on tori by relaxing the assumption on the spectrum. We don’t need the spectrum to be real anymore and allow pairs of complex eigenvalues. We can handle corresponding 2-dimensional intermediate foliations with the help of Continuous Amenable Reduction Theorem (due to Boris and Victoria). Interestingly there is also a partial generalization to certain partially hyperbolic toral automorphisms:

Theorem Let ${L : \mathbb T^d \rightarrow\mathbb T^d}$ be a totally irreducible automorphism with exactly two eigenvalues of modulus one and simple real eigenvalues away from the unit circle. Let ${f}$ be a volume-preserving ${C^{22}}$-small perturbation of ${L}$ such that the Lyapunov exponents of ${f}$ with respect to the volume are the same as the Lyapunov exponents of ${L}$. Then ${f}$ is ${C^{1+\varepsilon}}$ conjugate to ${L}$.

## Length Spectrum Rigidity

August 21, 2018

Here I would like to point a version local length spectrum rigidity for hyperbolic surfaces.

Let ${S}$ be a surface of genus ${\ge 2}$. If ${g}$ is negatively curved Riemannian metric on ${S}$ then each non-trivial homotopy class of loops on ${S}$ contains a unique unit speed geodesic representative. The collection of lengths with multiplicities of all such geodesics

$\displaystyle \ell_1\le\ell_2\le\ell_3\le\ldots$

is called the length spectrum of ${(S, g)}$ and we denote it by ${LS(g)}$.

Recall that ${(S, g_0)}$ is called hyperbolic if ${g_0}$ has constant curvature ${-1}$.

Proposition Let ${(S, g_0)}$ be a hyperbolic surface. Then there exists a ${C^2}$-neighborhood ${\mathcal U}$ of ${g_0}$ in the space of smooth Riemannian metrics such if ${g_1\in \mathcal U}$, ${LS(g_1)=LS(g_0)}$ and ${\textup{Area}(g_1)=\textup{Area}(g_0)=2\pi|\chi(S)|}$ then ${g_1}$ is isometric to ${g_0}$. That is, there exists a diffeomorphism ${\varphi\colon S\rightarrow S}$ such that ${g_1=\varphi^*g_0}$.

We give two proofs. The first one just combines Margulis’ asymptotics for the number of periodic orbits of length ${\le T}$ and Katok’s entropy rigidity theorem.

## Shenzhen conference photos

July 12, 2018

Some highlights of Shenzhen conference are under the fold.

## Shenzhen problems: zero entropy

July 11, 2018

These two questions were stated by me in Shenzhen after Federico Rodriguez Hertz’s talk on flexibility of Lyapunov exponents. I stumbled into these problems after discussions with Bryce Weaver, when we unsuccessfully tried to show that certain higher dimensional geodesic flows are stably non-uniformly hyperbolic. My last mathematical discussion in January 2018 with late Anatole Katok was revolving around these questions and he was leaning towards existence of counterexamples.

Question 1
Let ${S}$ be a surface of genus ${\ge 2}$ equipped with a smooth (at least ${C^{2+\varepsilon}}$) Riemannian (symmetric Finsler) metric. Is it true that the metric entropy of the geosedic flow with respect to the Liouville measure is positive?

Question 2
Let ${f\colon\mathbb T^2\rightarrow\mathbb T^2}$ be an area preserving smooth (at least ${C^2}$ or ${C^{1+\varepsilon}}$) diffeomorphism homotopic to 2-1-1-1 automorphism. Is it true that the metric entropy of ${f}$ with respect to the area is positive?

## Shenzhen problems

July 11, 2018

Below the fold I record open problems from my Shenzhen lectures. These may seem quite random, however they appeared naturally in the lectures.

## Transitivity of strong unstable foliation

May 12, 2018

This is to give an update on problem 3 from my 2009 post. Recently Aleksey Kolmogorov, Itai Maimon and I performed high precision computer experiments which confirmed that the strong unstable manifold through the fixed point $W^{uu}(p)$ is dense in $\mathbb T^3$. Based on our computer experiments we conjectured that the strong unstable foliation is transitive and that there is a unique invariant measure with absolutely continuous conditionals on strong unstable leaves ($u$-Gibbs measure), which is then, of course, the SRB measure. For example the image below shows the points of intersection of the strong unstable manifold of the fixed point and a $\mathbb T^2$ transversal for the Anosov diffeomorphism

$(x,y,z)\mapsto (2x+y+0.1\sin(2\pi x),x+2y+z,y+z)$

Our first conjecture was proved by Jana Rodriguez Hertz and Raul Ures in a paper posted on arXiv yesterday. In fact, they proved a more general statement.

Theorem. Let $L\colon\mathbb T^3\to\mathbb T^3$ be a hyperbolic automophism with real spectrum $0<\lambda^s<1<\lambda^{wu}<\lambda^{uu}$. Let $f$ be an Anosov diffeomorphism homotopic to A with a partially hyperbolic splitting of the unstable bundle into weak unstable and strong unstable subbundles. Then the strong unstable foliation $W^{uu}$ of $f$ is transitive.

Some ingredients in the proof come from earlier results of Rodriguez Hertz-Rodriguez Hertz-Ures, Didier and Ren-Gan-Zhang. Below the fold I record a self-contained proof following Rodriguez Hertz-Ures. Needless to say that all possible mistakes are due to me.

Note however that denseness of leaves $W^{uu}(p)$ through fixed or periodic points $p$ remains unknown as well as the minimality question.