# Polynomial log-volume growth in slow dynamics and the GK-dimensions of twisted homogeneous coordinate rings

@inproceedings{Lin2021PolynomialLG, title={Polynomial log-volume growth in slow dynamics and the GK-dimensions of twisted homogeneous coordinate rings}, author={Hsueh-Yung Lin and Keiji Oguiso and De-Qi Zhang}, year={2021} }

Twisted homogeneous coordinate rings are natural invariants associated to a projective variety X with an automorphism f . We study the Gelfand-Kirillov dimensions of these noncommutative algebras from the perspective of complex dynamics, by noticing that when X is a smooth complex projective variety, they essentially coincide with the polynomial logarithmic volume growth Plov(f) of (X, f). We formulate some basic dynamical properties about these invariants and study explicit examples. Our main…

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## References

SHOWING 1-10 OF 28 REFERENCES

### Automorphisms of compact Kähler manifolds with slow dynamics

- MathematicsTransactions of the American Mathematical Society
- 2020

We study the automorphisms of compact Kahler manifolds having slow dynamics. By adapting Gromov's classical argument, we give an upper bound on the polynomial entropy and study its possible values in…

### On the cohomological action of automorphisms of compact K\"ahler threefolds

- Mathematics
- 2017

Extending well-known results on surfaces, we give bounds on the cohomological action of automorphisms of compact K\"ahler threefolds. More precisely, if the action is virtually unipotent we prove…

### ON THE DYNAMICAL DEGREES OF MEROMORPHIC MAPS PRESERVING A FIBRATION

- Mathematics
- 2011

Let f be a dominant meromorphic self-map on a compact Kahler manifold X which preserves a meromorphic fibration π : X → Y of X over a compact Kahler manifold Y. We compute the dynamical degrees of f…

### CRITERIA FOR σ-AMPLENESS

- Mathematics
- 2000

In the past ten years a study of “noncommutative projective geometry” has flourished. By using and generalizing techniques of commutative projective geometry, one can study certain noncommutative…

### Noncommutative graded domains with quadratic growth

- Mathematics
- 1995

Letk be an algebraically closed field, and letR be a finitely generated, connected gradedk-algebra, which is a domain of Gelfand-Kirillov dimension two. Write the graded quotient ringQ(R) ofR…

### Suites d’Applications Méromorphes Multivaluées et Courants Laminaires

- Mathematics
- 2005

Let Fn: X1 → X2 be a sequence of (multivalued) meromorphic maps between compact Kähler manifolds. We study the asymptotic distribution of preimages of points by Fn and, for multivalued self-maps of a…

### Explicit Examples of rational and Calabi-Yau threefolds with primitive automorphisms of positive entropy

- Mathematics
- 2013

We present the first explicit examples of a rational threefold and a Calabi-Yau threefold, admitting biregular automorphisms of positive entropy not preserving any dominant rational maps to lower…

### Some aspects of explicit birational geometry inspired by complex dynamics

- Mathematics
- 2014

Our aim is to illustrate how one can effectively apply the basic ideas and notions of topological entropy and dynamical degrees, together with recent progress of minimal model theory in higher…

### Growth of Algebras and Gelfand-Kirillov Dimension

- Mathematics
- 1985

Introduction Growth of algebras Gelfand-Kirillov dimension of algebras Gelfand-Kirillov dimension of related algebras Localization Modules Graded and filtered algebras and modules Almost commutative…