Shenzhen conference photos

July 12, 2018

Some highlights of Shenzhen conference are under the fold.

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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.

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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.

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Anatole Katok

May 1, 2018

In this posting I will be collecting links to online materials.

  1. Anatole’s own page at PennState contains a lot of his writings, including expository ones such as papers on history of dynamics and mathematics in general.  In particular: preface to the 60th year old BD volume, which includes the 1960 Kronrod seminar picture;  1965 Humsan school;  tribute to Anosov; Women in Soviet Math (joint w Svetlana); 2014 MSRI photos;

  2. Moscow Math Journal on the occasion of Anatole’s 60th birthday
  3. Mathnet-ru page gives a good idea on Anatole’s work prior to emigration
  4. Pictures from 2004 conference on Svetlana’s page:
  5. E.B. Dynkin interviews with Anatole and Svetlana    ,,,,,
  6. BIRS group photo 2014:
  7. Oberwolfach photo collection:
  8. Kiev Mathematics Colloquium. Flexibility of entropies and Lyapunov exponents. Anatole and Svetlana visited Kiev in (May?) 2015 after the orange revolution to show support for Ukraine and Ukrainian Mathematics.
  9. Math genealogy seem to be pretty accurate with a few omissions (Satayev for example)
  10. Memorial Website:

Anatole Katok, 1944-2018

May 1, 2018

Anatole Katok,  August 9, 1944 — April 30, 2018


It is really hard to believe Anatole (Tolya) passed away yesterday.

He was born in Washington D.C. in August of 1944. His parents were Soviet citizens working in the american capital in relation with Lend-Lease Act. Then he was brought to the Soviet Union as a baby in 1945.  Mathematical career of Anatole began in Moscow in the 60s with work on periodic approximations in ergodic theory joint with A. Stepin. His advisor was Ya. G. Sinai and, later, Anatole was also very much influenced by D. V. Anosov. In 1978 Anatole and Svetlana emigrated to the United States (via Vienna and Rome if I remember correctly). Anatole went on to have a distinguished career in the US. He will be remembered for his mathematics, his personality, his “bible” of dynamical systems joint with B. Hasselblatt.  I am sure that once the shock subsides a lot more will be said about his life and his mathematics.

This picture reflects very well how I remember Anatole. R.I.P.


50 years of Berkeley encounter

March 20, 2018

So, if all goes well, such as me securing a Chinese visa, I will be participating in the International Conference on Dynamical Systems in Shenzhen, China, in June. The occasion is somewhat unusual and romantic: it is 50 years anniversary of Berkeley encounter.

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Rigidity of hyperbolic automorphisms from volume Lyapunov spectra

March 15, 2018

A recent preprint of Radu Saghin and Jiagang Yang is devoted to rigidity of hyperbolic automorphisms and partially hyperbolic skew products. The authors take a perspective similar to that of A. Katok’s entropy rigidity for geodesic flows on surfaces. Recall that by entropy rigidity the Lyapunov exponent of the Liouville measure achieves its maximum value for hyperbolic surfaces, when metrics are varied in the space of metrics of negative curvature with a fixed area. Hence the Lyapunov exponent determines the hyperbolic metric up to isometry and up to Teichmuller coordinate.

The model of Saghin-Yang is a hyperbolic automorphism. The most basic, and already very interesting, version of their result is the following.

Theorem 1 Let {L\colon\mathbb T^3\rightarrow\mathbb T^3} be a hyperbolic automorphism with distinct Lyapunov exponents

\displaystyle \chi^s<0<\chi^{wu}<\chi^{uu}

Assume that diffeomorphism {f} is a volume preserving {C^1}-perturbation of {L} and that the volume Lyapunov exponents stay the same:

\displaystyle \chi^s_f=\chi^s,\,\, \chi^{wu}_f=\chi^{wu},\,\, \chi^{uu}_f=\chi^{uu}

Then {f} is {C^\infty} conjugate to {L}.

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February 14, 2018

I learned about the {\mathcal A}-operation from Neretin’s book. The Continuum Hypothesis (CH) for Borel sets says that any Borel subset of the real line {\mathbb R} is either countable or has continuum cardinality. Indeed what a natural problem!

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Luzin and the Continuum Hypothesis

February 4, 2018

So I finished reading Yury Neretin’s “Nikolay Luzin, his students, adversaries, and defenders (notes on the history of Moscow mathematics, 1914-1936)” which is an impressive historical investigation around Luzin’s life and, of course, Luzin’s affair. 

Neretin performed a monumental combing through available sources and his manuscript is full of quotations. Regarding the affair per se Neretin holds an unorthodox view that Moscow mathematicians were at the origin of the attack on Luzin. This is at odds with Yushkewich-Demidov explanation that the affair was  launched by party apparatchik Kol’man with assistance of Editor in Chief of Pravda newspaper Mekhlis. In any case, I guess, this is no way of knowing for sure who wrote the anonymous Pravda article and it is not that important anyway. Read the rest of this entry »