Stochastic properties of dynamical systems

Abstract

This project concerns problems in the field of Dynamical Systems, in particular in connection with their stochastic properties. The project plans to address some outstanding open problems in the field of Dynamical Systems, e.g., the derivation of precise limit laws for smooth hyperbolic systems, the study of partially hyperbolic systems, discontinuous systems, extended systems, systems with many, possibly infinitely many, degrees of freedom, and systems with mixed phase space.The project plans to address some outstanding open problems in the field of Dynamical Systems, e.g., the derivation of precise limit laws for smooth hyperbolic systems, the study of partially hyperbolic systems, discontinuous systems, extended systems, systems with many, possibly infinitely many, degrees of freedom, and systems with mixed phase space.

Results achieved

: Description of the achievement of the objectives connected to the project and related outcomes The project followed a high-risk/high-gain approach, where intermediate results were expected to generate significant advancements independently of the full completion of long-term objective. The general Objective was to advance the theoretical and applied understanding of complex dynamical systems through the development of new analytical tools and the study of representative models. Specific Objectives and Outcomes (TO1) Dynamical zeta functions, Partially achieved (new results for parabolic systemsand escape rates) (TO2) Refined limit laws. Fully achieved (multiple results on CLT, EVT, Poisson laws) (TO3) Partially hyperbolic systems. Partially achieved (new limit theorems, ongoing work) (TO4) Parabolic/renormalizable systems. Partially achieved (ongoing work) (TO6) Mean-field systems and Vlasov limits Fully achieved (phase transitions,convergence and stability results) (NC1, 2) Birkhoff sums and infinite mixing Fully achieved (quantitative infinite mixing, extensions of ergodicity) (NC3) Horocycle flow and continued fractions Fully achieved (multiple results) (NC4) Thermodynamic formalism (infinite measure) Partially achieved (ongoing work) (NC5) Lorentz gas Partially achieved (transport and random walk models) (MP1,3) Mixed phase space and random billiards Fully achieved (convex billiards and sequential billiards) The project activities, being purely theoretical and computational, comply with the DNSH principle as they involve no environmentally harmful processes. Open Access principles are respected through the dissemination of results in open repositories (ArXiv, the standard repository for physics and mathematics) and open-access journals where possible. All activities were conducted in accordance with gender equality, generational inclusion, and equal opportunity principles, ensuring non-discriminatory participation and support for early-career researchers. Detailed description of activities carried out by each operational unit with a focus on the timeframe for their implementation WP1 – Transfer Operators and Spectral Theory (TO) Activities implemented: • Development of anisotropic functional spaces for transfer operators • Study of spectral properties in presence of discontinuities • Analysis of statistical laws and linear response • Use of thermodynamic formalism • Mean field systems Main results: • New anisotropic frameworks (Butterley–Kim) • Identification of essential spectrum due to discontinuities (Butterley,Canestrari, Castorrini) • Major advances in linear response theory and statistical laws (Canestrari, Galatolo) • Properties of self-consistent operators and mean field systems (Galatolo, Castorrini, Liverani) • Derivation of the Heat equation from a mechanical model (Canestrari, Liverani) • New limit theorems, thermodynamics formalism (Bonanno, Castorrini, Galatolo, Lenci, Liverani) Deliverables: • Peer-reviewed publications • Preprints KPIs achieved: • Number of publications: >10 • New theoretical frameworks introduced: 2+ WP2 – Non-Compact and Infinite Systems (NC) Activities implemented: • Study of infinite ergodic theory and mixing properties • Analysis of horocycle flows and continued fractions • Investigation of extended systems and transport Main results: • Quantitative infinite mixing results (Giulietti, Lenci, Canestrari) • Extensions of exact and K-mixing systems (Lenci) • Horocycle flows (Castorrini, Bonanno) • New connections between number theory and dynamics (Bonanno, Carminati) Deliverables: • Publications and preprints • Development of new analytical techniques KPIs achieved: • Publications: >8 • New models analyzed: ≥5 WP3 – Mixed Phase Space and Physical Models (MP) Activities implemented: • Study of billiards and entropy production • Analysis of random and sequential billiards • Billiard-related phenomena Main results: • Positive entropy in generic billiards (Del Magno) • Sequential Billiards (Liverani) Deliverables: • Publications in top-tier journals • Preprints on stochastic and physical systems KPIs achieved: • Publications: >4 • Cross-disciplinary outputs: physics + mathematics Publications and Preprints summary • Total published papers: 25 • Journals: Advances in Mathematics, Annali SNS Pisa, Communications in Mathematical Physics, Nonlinearity, Journal of Statistical Physics, etc. • Total preprints: 17 • Several currently under review in leading journals Additional Outputs • Editorial contributions (special issues) • Interdisciplinary collaborations All milestones were reached within the planned timeframe. No significant delays were recorded. Scientific Impact • Advancement of spectral theory in non-smooth settings • New results on statistical properties of complex systems • Significant contributions to mean-field dynamics • Foundations of non-equilibrium statistical physics Methodological Impact • Development of transferable analytical tools • Cross-fertilization between probability, analysis, and physics Long-term Impact • Opening new research directions (nonlinear transfer operators, stochastic dynamics) • Strengthening international collaborations

Project details

Unibo Team Leader: Marco Lenci

Unibo involved Department/s:
Dipartimento di Fisica e Astronomia "Augusto Righi"

Coordinator:
Università  degli Studi di ROMA Tor Vergata(Italy)

Total Unibo Contribution: Euro (EUR) 44.000,00
Project Duration in months: 24
Start Date: 28/09/2023
End Date: 28/02/2026