Dynamical Systems at ICTP

12/09/2024

The Claytor-Gilmer Fellowship, which annually awards $50,000, was established to further excellence in mathematics research and to help generate wider and sustained participation by Black mathematicians. Applications are being accepted until September 30.

Learn more and apply: https://ow.ly/VuGC50SSh7t

06/10/2023

Dynamical Systems Seminar TODAY

Yakov Pesin (Pennsylvania State University)

How big is the set of Birkhoff generic points?

Friday 6 October 2023, at 16:00 Italian time
Also online at https://zoom.us/j/95832470887?pwd=SlFZdW4wVTl1ek55Y2JiMTJUOXFSdz09

Abstract: Given a homeomorphism of a compact metric space and a continuous function, I will discuss the question of how big the set of Birkhoff generic points associated with the function is. This set plays an important role in studying dynamics of the map. To answer this question I will consider various ways of “measuring” a set: by using invariant measures, computing its Hausdorff dimension, and determining its Baire category. In the talk I will concentrate on the last characteristic — the Baire category — which gives some information on the topology of the set. This will be an introductory talk to the subject and it will be self-contained.

https://indico.ictp.it/event/10553/

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11/09/2023

Coming up on Wednesday, everyone welcome!

Morning (10:30 - 12:30) - Venue: Fibonacci lecture room (Galileo Guest House)   10:30 Jernej Činč (ICTP, Italy and University of Maribor, Slovenia)   Title: Lebesgue measure-preserving maps on one-dimensional manifolds   Abstract: In this talk I will survey recent advances in the st...

28/06/2023

The results of a great Italian/British/Nigerian cooperation in ICTP! With Douglas Coates, Stefano Luzzatto and Mubarak Muhammad!

We study a class of one-dimensional full branch maps admitting two indifferent fixed points as well as critical points and/or unbounded derivative. Under some mild assumptions we prove the existence of a unique invariant mixing absolutely continuous probability measure, study its rate of decay of co...

28/06/2023

🌟 Great news! 🌟

I am absolutely thrilled to share the incredible achievements of our very own IMM Batch 3 student from COMSATS University Lahore Campus Muhammad Umer, who has not only been admitted to the prestigious fully funded International Labex Bezout program in M2 but has also successfully defended his thesis! 🎉📚

Here are some words from him;

“I'm honored and overjoyed to announce that I have been selected for the International Labex Bezout Master's scholarship in M2, focusing on the captivating field of math and computer science. 🌍🎓

This remarkable journey would not have been possible without the unwavering support of the entire IMM family. During my toughest moments, you stood by me, providing continuous encouragement and guidance. I am truly grateful for each and every one of you. 🙏❤️

A heartfelt appreciation goes to my exceptional supervisor, Dr. Maurizio Parton, Dr. Carina and Dr Abdul Jawad. Their contributions have played a vital role in my success. Their mentorship and expertise have propelled me forward, shaping my path toward greatness. 🤝🔬

Lastly, I would like to express my deepest gratitude to Dr. Stefano Luzzatto from ICTP, and Prof. Dr. Sarfraz Ahmad from Comsats university Lahore. Their vision and dedication have created an extraordinary platform for Pakistani students. 🇵🇰🌍”

Let us join together in congratulating our outstanding student for this momentous achievement and wishing him a bright and prosperous future ahead. May their journey be filled with continued success, endless opportunities, and groundbreaking discoveries.

🌟🌈

22/05/2023

From the one one and only Khadim War

DYNAMICAL SYSTEMS SEMINAR

Tuesday 23 May, at 16:00

Venue: Luigi Stasi seminar room (Leonardo da Vinci Building)

Khadim War (IMPA, Rio de Janeiro)

Title: Proof of the Verjovsky Conjecture

Abstract: In the early 70's Alberto Verjovsky conjectured that every co-dimension one Anosov flow on a manifold of dimension at least four is topologically equivalent to a suspension over hyperbolic total automorphism. In particular in dimension four, the conjecture says that there exists only one topological equivalence class of Anosov flows. This problem can be interpreted as part of one of Smale's problems: Which manifolds can support a hyperbolic dynamical system? In this talk we prove that this conjecture is true. Anosov flows are central examples of chaotic systems. We will give a general overview of the different type of Anosov flows and show how our result gives an important step toward a more general classification. We will also discuss some topological obstructions for a manifold to support such a system.

https://indico.ictp.it/event/10376/

All are welcome to attend.

Abstract: In the early 70's Alberto Verjovsky conjectured that every co-dimension one Anosov flow on a manifold of dimension at least four is topologically equivalent to a suspension over hyperbolic total automorphism. In particular in dimension four, the conjecture says that there exists only o...

21/02/2023

Statistical stability of interval maps with critical points and singularities, by Jose F. Alves, Dalmi Gama, Stefano Luzzatto.

We prove strong statistical stability of a large class of one-dimensional maps which may have an arbitrary finite number of discontinuities and of non-degenerate critical points and/or singular points with infinite derivative, and satisfy some expansivity and bounded recurrence conditions. This generalizes known results for maps with critical points and bounded derivatives and in particular proves statistical stability of Lorenz-like maps with critical points and singularities studied in [S. Luzzatto and W. Tucker. Non-uniformly expanding dynamics in maps with singularities and criticalities. Inst. Hautes Etudes Sci. Publ. Math., (89):179-226, 1999]. We introduce a natural metric on the space of maps with discontinuities which does not seem to have been used in the literature before.

We prove strong statistical stability of a large class of one-dimensional maps which may have an arbitrary finite number of discontinuities and of non-degenerate critical points and/or singular points with infinite derivative, and satisfy some expansivity and bounded recurrence conditions. This gene...

13/02/2023

Congratulations to Hamza Ounesli for completing his first paper! Hamza proved that there exists C^1 uniformly expanding circle maps with invariant probability measures equivalent to Lebesgue even though they can have unbounded distortion and arbitrarily bad regularity of the derivative.

A very original approach to a very original and interesting question. No doubt this is but the first of many very original contributions to mathematics!

We prove that for a given modulus of continuity $ω$, there is a $C^1$ uniformly expanding map of the circle whose derivative has $ω$ as an optimal modulus of continuity and has an invariant probability measure equivalent to Lebesgue with a continuous density, in particular, generically, this famil...

21/10/2022

10/08/2022

Congratulations to David Ruelle who, together with Joel Lebowitz and Elliot Lieb, has been awarded the prestigious ICTP Dirac Medal for 2022.

ICTP has awarded its 2022 Dirac Medal to three distinguished physicists "for groundbreaking and mathematically rigorous contributions to the understanding of the statistical mechanics of classical and quantum physical systems".

07/07/2022

ICTP M A T H E M A T I C S S E M I N A R S 2022

Thursday 7 July, at 16:00 hrs CEST
Speaker: Damien Thomine (Université Paris-Sud)
Title: Probabilistic potential theory for dynamical systems

Abstract: Probabilistic potential theory relates Markov chains and harmonic functions. It offers the tools to compute, for instance, the probability that a random walk hits a specified target before another; or, thinking in terms of open systems, that the walk leaves the space at a specified point.

We adapt those results to dynamical systems. Our aim is to estimate, in a spatially periodic hyperbolic system, the probability of hitting a specified target before another. This work involves tools from potential theory (specifically, a balayage identity), transfer operators with spectral degeneracies, and perturbative arguments.

This will be a hybrid seminar. All are very welcome to join either online or in person.
Venue: Luigi Stasi Lecture Room (ICTP Leonardo Da Vinci Building), for those wishing to attend in person.
Register in advance for this meeting:

https://zoom.us/meeting/register/tJIlc-ysqDooHNBsuerJh6gHICViOSjgR8YL

After registering, you will receive a confirmation email containing information about joining the meeting.

Abstract: Probabilistic potential theory relates Markov chains and harmonic functions. It offers the tools to compute, for instance, the probability that a random walk hits a specified target before another; or, thinking in terms of open systems, that the walk leaves the space at a specified point.....