Topological invariants of dynamical systems and spaces of holomorphic maps. I.

*(English)*Zbl 1160.37322The author develops a conceptual framework for the study of certain geometric objects from the viewpoint of dynamical systems, on the one hand, and applies then this novel approach to analyze geometric properties of spaces of holomorphic maps as well as spaces of complex subvarieties of Hermitean manifolds in the sequel. Starting from an arbitrary category of geometric spaces, the dynamic setup is as follows:

Given a space \(\underline X\) and a group \(\Gamma\), the space \(X:=\underline X^\Gamma\) of \(\underline X\)-valued maps on \(\Gamma\) comes with a natural left \(\Gamma\)-action and is called the full shift space over \(\Gamma\) with the alphabet \(\underline X\). Finite subsets \(D\) of the group give rise to particularly interesting subspaces of \(\underline X^\Gamma\), the so-called subshifts of finite type. Roughly speaking, the author’s chain goal is to extend certain inariants, properties, or even theories in the given base category of geometric spaces to a suitable class of shift spaces over \(\Gamma\), including full shirt spaces and subshifts of finite type, and that in such a way that the new \(\Gamma\)-invariants satisfy certain “dynamical” properties. This is done, in the first part of the paper, by defining the concept of “mean dimension” in various categories of \(\Gamma\)-spaces with respect to a so-called amenable group \(\Gamma\) [cf. D. S. Ornstein and B. Weiss, J. Anal. Math. 48, 1–141 (1987; Zbl 0637.28015)], and by analyzing the properties of this topological invariant of the associated dynamical system in great depth. The evaluation of the mean dimension for subshifts of finite type appears to be a particularly subtle task. The whole second section of the paper is devoted to exhibiting a large variety of concrete geometric exarnples, where the mean dimension of a subshift \(Y\subset\underline X^\Gamma\) can be effectively determined in terms of \(\dim\underline X\) and the number of difference equations defining \(Y\).

The most interesting spaces in this context appear as solutions of elliptic differential equations over manifolds \(V\) with group actions. This includes harmonic maps \(V\to\underline X\) between Riemannian manifolds, where \(V\) is non-compact, \(\underline X\) is compact, and where \(V\) comes along with an isometry group \(\Gamma\) such that \(V/\Gamma\) is oompact. Another class of examples is provided by spaces of subvarieties in Riemannian manifolds. More precisely, the space \(X\) of all closed subsets of a Riemannian manifold \(W\) is a compact space with respect to the Hausdorff convergence topology on compact parts of \(W\) and any isometry group \(\Gamma\) of \(W\) acts continuously on the space \(X\).

In these special situations, the author proves more refined results on the behaviour of the mean dimension. Namely, in the third section of the paper, effective bounds for the mean dimension of spaces of holomorphic maps from Hermitean manifolds to complex projective spaces are exhibited, together with striking applications to minimal varieties, embedding problems, and rational curves in projective algebraic varieties, respectively. Spaces of \(n\)-dimensional complex subvarieties of a given Hermiteam manifold \(W\) acted upon by a cocompact amenable group \(\Gamma\) of isometries are then studied in the last section of the paper. The main results, in this context, concern lower and upper bounds for the mean dimension of such spaces as well as their continuity behaviour.

All together, the author’s approach provides a highly enlightening link between geometry and the theory of dynamical systems. Both his new conceptual framework and his refined results are of absolutely pioneering character in contemporary mathematics, and a great inspiration for further research likewise. Apart from the wealth of new results and related conjectures given in the current paper, it is the vast amount of illustrating examples that demonstrates the seemingly enormous power of the author’s dynamical viewpoint in different branches of geometric dimension theory and in value distribution theory.

Given a space \(\underline X\) and a group \(\Gamma\), the space \(X:=\underline X^\Gamma\) of \(\underline X\)-valued maps on \(\Gamma\) comes with a natural left \(\Gamma\)-action and is called the full shift space over \(\Gamma\) with the alphabet \(\underline X\). Finite subsets \(D\) of the group give rise to particularly interesting subspaces of \(\underline X^\Gamma\), the so-called subshifts of finite type. Roughly speaking, the author’s chain goal is to extend certain inariants, properties, or even theories in the given base category of geometric spaces to a suitable class of shift spaces over \(\Gamma\), including full shirt spaces and subshifts of finite type, and that in such a way that the new \(\Gamma\)-invariants satisfy certain “dynamical” properties. This is done, in the first part of the paper, by defining the concept of “mean dimension” in various categories of \(\Gamma\)-spaces with respect to a so-called amenable group \(\Gamma\) [cf. D. S. Ornstein and B. Weiss, J. Anal. Math. 48, 1–141 (1987; Zbl 0637.28015)], and by analyzing the properties of this topological invariant of the associated dynamical system in great depth. The evaluation of the mean dimension for subshifts of finite type appears to be a particularly subtle task. The whole second section of the paper is devoted to exhibiting a large variety of concrete geometric exarnples, where the mean dimension of a subshift \(Y\subset\underline X^\Gamma\) can be effectively determined in terms of \(\dim\underline X\) and the number of difference equations defining \(Y\).

The most interesting spaces in this context appear as solutions of elliptic differential equations over manifolds \(V\) with group actions. This includes harmonic maps \(V\to\underline X\) between Riemannian manifolds, where \(V\) is non-compact, \(\underline X\) is compact, and where \(V\) comes along with an isometry group \(\Gamma\) such that \(V/\Gamma\) is oompact. Another class of examples is provided by spaces of subvarieties in Riemannian manifolds. More precisely, the space \(X\) of all closed subsets of a Riemannian manifold \(W\) is a compact space with respect to the Hausdorff convergence topology on compact parts of \(W\) and any isometry group \(\Gamma\) of \(W\) acts continuously on the space \(X\).

In these special situations, the author proves more refined results on the behaviour of the mean dimension. Namely, in the third section of the paper, effective bounds for the mean dimension of spaces of holomorphic maps from Hermitean manifolds to complex projective spaces are exhibited, together with striking applications to minimal varieties, embedding problems, and rational curves in projective algebraic varieties, respectively. Spaces of \(n\)-dimensional complex subvarieties of a given Hermiteam manifold \(W\) acted upon by a cocompact amenable group \(\Gamma\) of isometries are then studied in the last section of the paper. The main results, in this context, concern lower and upper bounds for the mean dimension of such spaces as well as their continuity behaviour.

All together, the author’s approach provides a highly enlightening link between geometry and the theory of dynamical systems. Both his new conceptual framework and his refined results are of absolutely pioneering character in contemporary mathematics, and a great inspiration for further research likewise. Apart from the wealth of new results and related conjectures given in the current paper, it is the vast amount of illustrating examples that demonstrates the seemingly enormous power of the author’s dynamical viewpoint in different branches of geometric dimension theory and in value distribution theory.

Reviewer: Werner Kleinert (Berlin)

##### MSC:

37B99 | Topological dynamics |

32H02 | Holomorphic mappings, (holomorphic) embeddings and related questions in several complex variables |

58E20 | Harmonic maps, etc. |

53C23 | Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces |

32H30 | Value distribution theory in higher dimensions |

54H20 | Topological dynamics (MSC2010) |