Series of talks today, 21 July - Stasi Room at 15:00

[ICTP Math Section]

*TODAY - 21 July, 15:00 - 17:00*
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*Stasi Room, ICTP Leonardo Building*
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*15:00 - 15:30 Marks Ruziboev, **National Pedagogical University of 
Uzbekistan *
/On the Dynamics of Mean-Field Coupled Interval Maps with Singularities/
*Abstract: *In this talk, we present some results for mean-field coupled 
interval maps with finitely many singularities in the thermodynamic 
limit. Therefore,  the dynamics is governed by a nonlinear, 
self-consistent transfer operator.  We show that when the individual 
maps have good spectral properties, the coupled system also admits a 
unique equilibrium, which is exponentially stable.
*15:40 -16:10 Alejandro Passeggi, **Universidad de la República, Uruguay*
/Topological Criteria for Annular Chaos /
*Abstract: *Although paradigmatic models of chaotic dynamics in 
low-dimensional systems are well understood, proving that a given system 
exhibits chaotic behavior often remains a challenging task. Moreover, 
identifying the underlying mechanisms responsible for such dynamics is 
frequently beyond the scope of the classical literature on the subject.
In recent years, several topological criteria have been established for 
systems whose Poincaré map is defined on the annulus. These criteria 
provide simple and robust conditions guaranteeing the existence of chaos 
in the form of a rotational horseshoe. Roughly speaking, it is enough to 
find two topological disks with different rotation behavior under one 
iteration and whose forward iterates visit each other. This approach 
yields rigorous proofs of chaotic dynamics while relying on elementary 
information about the system [1,2]. Furthermore, effective 
implementations of these criteria have led to several concrete 
applications [3,4].
In this talk, I will review these results and discuss recent progress 
toward a natural next step: obtaining explicit constructions of the 
rotational horseshoe once the above criteria (or related ones) have been 
verified. Such constructions not only yield a rigorous computation of 
the map’s topological entropy, but also allow one to locate the 
rotational horseshoe and its associated essential instability region.
[1] A. Passeggi and F. A. Tal, Conditions Implying Annular Chaos, 
accepted to Inventiones Mathematicae.
[2] A. Passeggi and F. Pirán, Annular Chaos for Non-Wandering 
Homeomorphisms, arXiv.
[3] M. J. Capiński, M. Gröger, A. Passeggi and F. A. Tal, Conditions 
Implying Annular Chaos: Qualitative Results and CAP, arXiv.
[4] M. J. Capiński, S. Llavayol and A. Passeggi, Rotational Chaos in the 
Driven Pendulum (to appear).
*16:30 -17:00 Ali Tahzibi, **Universidade de São Paulo, Brazil*
/Stably exponentially mixing endomorphisms/
In a joint work with M. Hedyehloo, M. Nassiri, and H. Rajabzadeh, we 
obtain new examples of stable exponentially mixing endomorphisms beyond 
the uniformly expanding setting.
More precisely, we obtain two distinct C1 open classes of maps:
1. A set U of endomorphisms that are neither uniformly expanding nor 
admitting any dominated splitting, yet every smooth map in U has an ACIP 
and is exponentially mixing,
2. Another set V of endomorphisms that are not uniformly hyperbolic but 
do admit dominated splitting, and every smooth map in V has an ACIP and 
is exponentially mixing.
These sets are in the isotopy class of arbitrary uniformly expanding 
maps. The results are in the framework of virtually expanding 
endomorphisms introduced by M. Tsujii.

All are welcome to join.

-- 
Micol Stock
Secretariat of the ICTP Math Section

Abdus Salam International Centre for Theoretical Physics
Strada Costiera 11
34151 Trieste
Italy

Tel.: (+39) 040 2240 4455