We give an example of a dominant rational self-map of the projective plane whose dynamical degree is a transcendental number.

Data-based detection and quantification of causation in complex, nonlinear dynamical systems is of paramount importance to science, engineering, and beyond. Inspired by the widely used methodology in recent years, the cross-map-based techniques, we develop a general framework to advance towards a comprehensive understanding of dynamical causal mechanisms, which is consistent with the natural interpretation of causality. In particular, instead of measuring the smoothness of the cross-map as conventionally implemented, we define causation through measuring the scaling law for the continuity of the investigated dynamical system directly. The uncovered scaling law enables accurate, reliable, and efficient detection of causation and assessment of its strength in general complex dynamical systems, outperforming those existing representative methods. The continuity scaling-based framework is rigorously established and demonstrated using datasets from model complex systems and the real world.

We show that for any amenable group \Gamma and any Z\Gamma-module M of type FL with vanishing Euler characteristic, the entropy of the natural \Gamma-action on the Pontryagin dual of M is equal to the L^2-torsion of M. As a particular case, the entropy of the principal algebraic action associated with the module Z\Gamma / Z\Gamma f is equal to the logarithm of the Fuglede-Kadison determinant of f whenever f is a non-zero-divisor in Z\Gamma. This confirms a conjecture of Deninger. As a key step in the proof we provide a general Szeg\H{o}-type approximation theorem for the Fuglede-Kadison determinant on the group von Neumann algebra of an amenable group. As a consequence of the equality between L^2-torsion and entropy, we show that the L^2-torsion of a non-trivial amenable group with finite classifying space vanishes. This was conjectured by Lueck. Finally, we establish a Milnor-Turaev formula for the L^2-torsion of a finite \Delta-acyclic chain complex.

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In this paper we discuss properties of the KdV equation under periodic boundary conditions, especially those which are important to study perturbations of the equation. Next we review what is known now about long-time behaviour of solutions for perturbed KdV equations.