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. 
 

 

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).
 

 

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.