Theory of stochastic transitions in area preserving maps

A famous aspect of discrete dynamical systems defined by area-preserving maps is the physical interpretation of stochastic transitions occurring locally which manifest themselves through the destruction of invariant KAM curves and the local or global onset of chaos. Despite numerous previous investigations (see in particular Chirikov, Greene, Percival, Escande and Doveil and MacKay) based on different approaches, several aspects of the phenomenon still escape a complete understanding and a rigorous description. In particular Greene's approach is based on several conjectures, one of which is that the stochastic transition leading to the destruction of the last KAM curve in the standard map is due the linear destabilization of the elliptic points belonging to a peculiar family of invariants sets {I(m,n)} (rational iterates) having rational winding numbers and associated to the last KAM curve. Purpose of this work is to analyze the nonlinear phenomena leading to the stochastic transition in the standard map and their effect on the destabilization of the invariant sets associated to the KAM curves, leading, ultimately, to the destruction of the KAM curves themselves.Comment: 6 pages, 1 figure. Contributed to the Proceedings of the 24th International Symposium on Rarefied Gas Dynamics, July 10-16, 2004 Porto Giardino Monopoli (Bari), Ital

Holographic screens in ultraviolet self-complete quantum gravity

In this paper we study the geometry and the thermodynamics of a holographic screen in the framework of the ultraviolet self-complete quantum gravity. To achieve this goal we construct a new static, neutral, non-rotating black hole metric, whose outer (event) horizon coincides with the surface of the screen. The space-time admits an extremal configuration corresponding to the minimal holographic screen and having both mass and radius equalling the Planck units. We identify this object as the space-time fundamental building block, whose interior is physically unaccessible and cannot be probed even during the Hawking evaporation terminal phase. In agreement with the holographic principle, relevant processes take place on the screen surface. The area quantization leads to a discrete mass spectrum. An analysis of the entropy shows that the minimal holographic screen can store only one byte of information while in the thermodynamic limit the area law is corrected by a logarithmic term.Comment: 18 pages, 4 figures; v2 additional references; v3 shortened version in press as invited contribution to "Black hole Physics'', special issue of Advances of High Energy Physics edited by X. Zeng, C. Corda and D. Che

Un-spectral dimension and quantum spacetime phases

In this Letter, we propose a new scenario emerging from the conjectured presence of a minimal length $\ell$ in the spacetime fabric, on the one side, and the existence of a new scale invariant, continuous mass spectrum, of un-particles on the other side. We introduce the concept of \textit{un-spectral dimension} $\mathbb{D}_U$ of a $d$-dimensional, euclidean (quantum) spacetime, as the spectral dimension measured by an "un-particle" probe. We find a general expression for the un-spectral dimension $\mathbb{D}_U$ labelling different spacetime phases: a semi-classical phase, where ordinary spectral dimension gets contribution from the scaling dimension $d_U$ of the un-particle probe ; a critical "Planckian phase", where four-dimensional spacetime can be effectively considered two-dimensional when $d_U=1$; a "Trans-Planckian phase", which is accessible to un-particle probes only, where spacetime as we currently understand it looses its physical meaning.Comment: 5 pages, 1 figure, version matching that published by Physics Letters

The Hawking-Page crossover in noncommutative anti-deSitter space

We study the problem of a Schwarzschild-anti-deSitter black hole in a noncommutative geometry framework, thought to be an effective description of quantum-gravitational spacetime. As a first step we derive the noncommutative geometry inspired Schwarzschild-anti-deSitter solution. After studying the horizon structure, we find that the curvature singularity is smeared out by the noncommutative fluctuations. On the thermodynamics side, we show that the black hole temperature, instead of a divergent behavior at small scales, admits a maximum value. This fact implies an extension of the Hawking-Page transition into a van der Waals-like phase diagram, with a critical point at a critical cosmological constant size in Plank units and a smooth crossover thereafter. We speculate that, in the gauge-string dictionary, this corresponds to the confinement "critical point" in number of colors at finite number of flavors, a highly non-trivial parameter that can be determined through lattice simulations.Comment: 24 pages, 6 figure, 1 table, version matching that published on JHE