scholarly journals Spectral interpretations of dynamical degrees and applications

2021 ◽  
Vol 194 (1) ◽  
pp. 299
Author(s):  
Dang ◽  
Favre
Keyword(s):  
Dynamical Degrees

scholarly journals Dynamical degrees of affine-triangular automorphisms of affine spaces

2021 ◽  
pp. 1-42
Author(s):  
JÉRÉMY BLANC ◽  
IMMANUEL VAN SANTEN
Keyword(s):  
Affine Space ◽  
Affine Spaces ◽  
Dynamical Degree ◽  
Perron Number ◽  
Dynamical Degrees ◽  
Affine Automorphism

Abstract We study the possible dynamical degrees of automorphisms of the affine space $\mathbb {A}^n$ . In dimension $n=3$ , we determine all dynamical degrees arising from the composition of an affine automorphism with a triangular one. This generalizes the easier case of shift-like automorphisms which can be studied in any dimension. We also prove that each weak Perron number is the dynamical degree of an affine-triangular automorphism of the affine space $\mathbb {A}^n$ for some n, and we give the best possible n for quadratic integers, which is either $3$ or $4$ .

ENTROPY OF A NONSTATIC BLACK HOLE

2000 ◽  
Vol 15 (28) ◽  
pp. 1739-1747 ◽  
Author(s):  
LI XIANG ◽  
ZHAO ZHENG
Keyword(s):  
Black Hole ◽  
Black Hole Entropy ◽  
Time Dependent ◽  
Brick Wall ◽  
Two Dimensional ◽  
De Sitter ◽  
Wall Model ◽  
New Model ◽  
Brick Wall Model ◽  
Dynamical Degrees

We point out that the brick-wall model cannot be applied to the nonstatic black hole. In the case of a static hole, we propose a new model where the black hole entropy is attributed to the dynamical degrees of the field covering the two-dimensional membrane just outside the horizon. A cutoff different from the model of 't Hooft is necessarily introduced. It can be treated as an increase in horizon because of the space–time fluctuations. We also apply our model to the nonequilibrium and nonstatic cases, such as Schwarzschild–de Sitter and Vaidya space–times. In the nonstatic case, the entropy relies on a time-dependent cutoff.

scholarly journals SOLUTION TO 2+1 GRAVITY IN DREIBEIN FORMALISM

1994 ◽  
Vol 03 (01) ◽  
pp. 131-137 ◽  
Author(s):  
W.G. UNRUH
Keyword(s):  
General Relativity ◽  
Degrees Of Freedom ◽  
Closed Timelike Curves ◽  
Original Theory ◽  
Arbitrary Genus ◽  
Genus 2 ◽  
Genus Two ◽  
Dynamical Degrees

This paper outlines the reduction of the dreibein formalism of 2+1 General Relativity to the dynamical degrees of freedom for a genus 2 (and by extension for an arbitrary genus) two space. The resulting dynamical variables of the reduced theory are global holonomies and are constants of the motion of the original theory. The relation to geometry and closed timelike curves is briefly described.

scholarly journals Canonical Heights on Hyper-Kähler Varieties and the Kawaguchi–Silverman Conjecture

2019 ◽  
Author(s):  
John Lesieutre ◽  
Matthew Satriano
Keyword(s):  
Projective Variety ◽  
Kähler Manifold ◽  
Height Function ◽  
Kähler Manifolds ◽  
Kahler Manifolds ◽  
Dense Orbit ◽  
Canonical Height ◽  
Projective Threefolds ◽  
Higher Dimensional ◽  
Dynamical Degrees

Abstract The Kawaguchi–Silverman conjecture predicts that if $f: X \dashrightarrow X$ is a dominant rational-self map of a projective variety over $\overline{{\mathbb{Q}}}$, and $P$ is a $\overline{{\mathbb{Q}}}$-point of $X$ with a Zariski dense orbit, then the dynamical and arithmetic degrees of $f$ coincide: $\lambda _1(f) = \alpha _f(P)$. We prove this conjecture in several higher-dimensional settings, including all endomorphisms of non-uniruled smooth projective threefolds with degree larger than $1$, and all endomorphisms of hyper-Kähler manifolds in any dimension. In the latter case, we construct a canonical height function associated with any automorphism $f: X \to X$ of a hyper-Kähler manifold defined over $\overline{{\mathbb{Q}}}$. We additionally obtain results on the periodic subvarieties of automorphisms for which the dynamical degrees are as large as possible subject to log concavity.

Dynamical Degrees of Freedom of DNA Sequences by Local and Global Short-Range Prediction

Fractals ◽  
1997 ◽  
Vol 05 (01) ◽  
pp. 1-10
Author(s):  
M. Ragosta ◽  
C. Serio ◽  
M. T. Lanfredi ◽  
M. Macchiato
Keyword(s):  
Dna Sequences ◽  
Degrees Of Freedom ◽  
Nonlinear Process ◽  
Short Term ◽  
Chaotic Motions ◽  
Deterministic Systems ◽  
Predictive Skill ◽  
Low Dimensional ◽  
Dynamical Degrees ◽  
Insight Into

The dynamical properties of DNA sequence samples have been analyzed on the basis of a procedure able to distinguish chaos from randomness. The procedure relies on the concept of short-term (range) predictability of low-dimensional chaotic motions and can distinguish merely linear stochastic processes, e.g. fractional Brownian motion, from truly nonlinear deterministic systems. The method consists in obtaining forecasts on the basis of past events in the sequence. Two forecasting strategies are used. The local strategy views the sequence as the outcome of a nonlinear process, whereas the global approach considers the series as the outcome of a linear stochastic process. For both approaches, the predictive skill is computed and their inter-comparison allows us to get insight into and an understanding of the structure of DNA sequences. Nucleotidic sequences belonging to different taxonomic and functional groups have been analyzed. Different behaviors have been detected according to the existence of finite correlation dimension for specific groups of sequences.

scholarly journals FROM BRICKS TO QUASINORMAL MODES: A NEW PERSPECTIVE ON BLACK HOLE ENTROPY

2013 ◽  
Vol 22 (12) ◽  
pp. 1342027 ◽  
Author(s):  
MICHELE ARZANO ◽  
STEFANO BIANCO ◽  
OLAF DREYER
Keyword(s):  
Black Hole ◽  
Short Distance ◽  
Degrees Of Freedom ◽  
Quasinormal Modes ◽  
Black Hole Entropy ◽  
Quantum Field ◽  
Extended Region ◽  
New Perspective ◽  
The Right ◽  
Dynamical Degrees

Calculations of black hole entropy based on the counting of modes of a quantum field propagating in a Schwarzschild background need to be regularized in the vicinity of the horizon. To obtain the Bekenstein–Hawking result, the short distance cut-off needs to be fixed by hand. In this note, we give an argument for obtaining this cut-off in a natural fashion. We do this by modeling the black hole by its set of quasinormal modes (QNMs). The horizon then becomes a extended region: the quantum ergosphere. The interaction of the quantum ergosphere and the quantum field provides a natural regularization mechanism. The width of the quantum ergosphere provides the right cut-off for the entropy calculation. We arrive at a dual picture of black hole entropy. The entropy of the black hole is given both by the entropy of the quantum field in the bulk and the dynamical degrees of freedom on the horizon.

scholarly journals Examples of rational maps of $CP^2$ with equal dynamical degrees and no invariant foliation

2016 ◽  
Vol 144 (2) ◽  
pp. 279-297
Author(s):  
Scott R. Kaschner ◽  
Rodrigo A. Pérez ◽  
Roland K. W. Roeder
Keyword(s):  
Rational Maps ◽  
Invariant Foliation ◽  
Dynamical Degrees

scholarly journals Dynamical invariants of monomial correspondences

2020 ◽  
pp. 1-16
Author(s):  
NGUYEN-BAC DANG ◽  
ROHINI RAMADAS
Keyword(s):  
Exponential Growth ◽  
Toric Varieties ◽  
Arithmetic Geometry ◽  
Degree Sequences ◽  
Asymptotic Rate ◽  
Dynamical Invariants ◽  
Dynamical Degrees

We focus on various dynamical invariants associated to monomial correspondences on toric varieties, using algebraic and arithmetic geometry. We find a formula for their dynamical degrees, relate the exponential growth of the degree sequences to a strict log-concavity condition on the dynamical degrees and compute the asymptotic rate of the growth of heights of points of such correspondences.

scholarly journals TOWERS OF GRAVITATIONAL THEORIES

2006 ◽  
Vol 15 (12) ◽  
pp. 2293-2302 ◽  
Author(s):  
WALTER D. GOLDBERGER ◽  
IRA Z. ROTHSTEIN
Keyword(s):  
Black Hole ◽  
Bound States ◽  
Degrees Of Freedom ◽  
Mechanical Energy ◽  
Point Of View ◽  
Dissipative Effects ◽  
Physical Measurements ◽  
Gravitational Theories ◽  
Theories Of Gravity ◽  
Dynamical Degrees

In this essay, we introduce a theoretical framework designed to describe black hole dynamics. The difficulties in understanding such dynamics stems from the proliferation of scales involved when one attempts to simultaneously describe all of the relevant dynamical degrees of freedom. These range from the modes that describe the black hole horizon, which are responsible for dissipative effects, to the long wavelength gravitational radiation that drains mechanical energy from macroscopic black hole bound states. We approach the problem from a Wilsonian point of view, by building a tower of theories of gravity each of which is valid at different scales. The methodology leads to multiple new results in diverse topics including phase transitions of Kaluza–Klein black holes and the interactions of spinning black hole in non-relativistic orbits. Moreover, our methods tie together speculative ideas regarding dualities for black hole horizons to real physical measurements in gravitational wave detectors.

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