## LxDS Spring School 2023

Workshop

24-26 May 2023, ISEG-Lisbon School of Economics & Management, Universidade de Lisboa

### Description

The LxDS-Lisbon Dynamical Systems group, the Mathematics Department of ISEG, CEMAPRE, REM and CMAFcIO are organizing a 3-day spring school (24th to 26th May 2023) on dynamical systems to be held at ISEG/ULisboa. The school will consist of three mini-courses in specific areas of dynamical systems lectured by specialists of recognized international merit.

REGISTRATION is required to confirm participation. Please fill the form on this web page.

Those wishing to give a 20-minute talk should mention it in the comments section of the registration form, indicating the corresponding title and abstract.

The spring school will have a limited number of participants.
PhD students can apply for financial support.

### Speakers:

Josef Hofbauer (University Vienna), Nicolas Chevallier (Université de Haute Alsace), Silvius Klein (PUC-Rio).

### Organizers:

Pedro Duarte (Universidade de Lisboa), José Pedro Gaivão (Universidade de Lisboa), João Lopes Dias (Universidade de Lisboa) and Telmo Peixe (Universidade de Lisboa)

### Outline

#### Diophantine approximations and the space of lattices (Nicolas Chevallier)

The objective of the course is to show the fruitful relations between  Diophantine approximations,  lattices and the action of the diagonal flow in the space of  unimodular lattices. We will start with the classics: continued fractions in one dimension, Dirichlets theorem in one dimension  and Dirichlets theorem in simultaneous approximation. Then we will introduce lattices, the space of unimodular lattices, the shortest vector function and the diagonal flow. We will reinterpret the notions of badly approximable and singular vectors (Dani correspondence). We will also reinterpret the notion of best Diophantine approximation in terms of minimal vectors in lattices. Finally we will give applications  of the ergodicity of the diagonal flow to Diophantine approximations.

#### Statistical properties for certain dynamical systems (Silvius Klein)

The main objective of this mini-course is the study of some recent topics in ergodic theory about limit laws (i.e. the large deviations principle and the central limit theorem) for certain types of dynamical systems. The main tool in this study is the existence of the spectral gap of the Markov transition operator or the Ruelle transfer operator in an appropriate space of observables.

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### Schedule

Time24th May25th May26th May
09h30Registration
10h00 - 11h30Course 1
(Part 1)
Course 1
(Part 2)
Course 1
(Part 3)
11h30 - 11h45Coffee breakCoffee breakCoffee break
11h45 - 13h15Course 2
(Part 1)
Course 2
(Part 2)
Course 2
(Part 3)
13h15 - 15h00LunchLunchLunch
15h00 - 16h30Course 3
(Part 1)
Course 3
(Part 2)
Course 3
(Part 3)
16h30 - 17h00Coffee breakCoffee break
17h00 - 17h30talk 1talk 3
17h30 - 18h00talk 2talk 4
20h00

### Target audience

PhD students, post-docs and researchers.

### Observations

The organizers are grateful for the sponsorship of the CIM - Centro Internacional de Matemática.

### Contacts

CEMAPRE - Centre for Applied Mathematics and Economics

Rua do Quelhas, n.º 6
1200-781 Lisboa
Portugal

Email: cemapre@iseg.ulisboa.pt
Tel: (+351) 213 925 876

### Registration

After registration you will be contacted by the event organization.