Let f be a dominant meromorphic self-map on a compact Kähler manifold X which preserves a meromorphic fibration π : X → Y of X over a compact Kähler manifold Y . We compute the dynamical degrees of f in terms of its dynamical degrees relative to the fibration and the dynamical degrees of the map g : Y → Y induced by f . We derive from this result new properties of some fibrations intrinsically associated to X when this manifold admits an interesting dynamical system.

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In this paper, we provide new results and algorithms (including backtracking versions of Nesterov accelerated gradient and Momentum) which are more applicable to large scale optimisation as in Deep Neural Networks. We also demonstrate that Backtracking Gradient Descent (Backtracking GD) can obtain good upper bound estimates for local Lipschitz constants for the gradient, and that the convergence rate of Backtracking GD is similar to that in classical work of Armijo. Experiments with datasets CIFAR10 and CIFAR100 on various popular architectures verify a heuristic argument that Backtracking GD stabilises to a finite union of sequences constructed from Standard GD for the mini-batch practice, and show that our new algorithms (while automatically fine tuning learning rates) perform better than current state-of-the-art methods such as Adam, Adagrad, Adadelta, RMSProp, Momentum and Nesterov accelerated gradient. To help readers avoiding the confusion between heuristics and more rigorously justified algorithms, we also provide a review of the current state of convergence results for gradient descent methods. Accompanying source codes are available on GitHub.

Let {\mathbb{K}} be an algebraically closed field of arbitrary characteristic, X and Y irreducible possibly singular algebraic varieties over {\mathbb{K}}. Let {f:X\vdash X} and {g:Y\vdash Y} be dominant correspondences, and {\pi:X\dashrightarrow Y} a dominant rational map which semi-conjugate f and g, i.e. so that {\pi\circ f=g\circ\pi}. We define relative dynamical degrees {\lambda_{p}(f|\pi)\geq 1} for any {p=0,\dots,\dim(X)-\dim(Y)}. These degrees measure the relative growth of positive algebraic cycles, satisfy a product formula when Y is smooth and g is a multiple of a rational map, and are birational invariants. More generally, a weaker product formula is proven for more general semi-conjugacies, and for any generically finite semi-conjugacy {(\varphi,\psi)} from {\pi_{2}:(X_{2},f_{2})\rightarrow(Y_{2},g_{2})} to {\pi_{1}:(X_{1},f_{1})\rightarrow(Y_{1},g_{1})} we have {\lambda_{p}(f_{1}|\pi_{1})\geq\lambda_{p}(f_{2}|\pi_{2})} for all p. Many of our results are new even when {\mathbb{K}=\mathbb{C}}. Self-correspondences are abundant, even on varieties having not many self rational maps, hence these results can be applied in many situations. In the last section of the paper, we will discuss recent new applications of this to algebraic dynamics, in particular to pullback on l-adic cohomology groups in positive characteristics.