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Convex Analysisの二,三の進展について

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Introduction to the Modern Theory of Dynamical Systems

Computational geometry

Convex Analysis: (pms-28)

Part I. Examples and Fundamental Concepts Introduction 1. First examples 2. Equivalence, classification, and invariants 3. Principle classes of asymptotic invariants 4. Statistical behavior of the orbits and introduction to ergodic theory 5. Smooth invariant measures and more examples Part II. Local Analysis and Orbit Growth 6. Local hyperbolic theory and its applications 7. Transversality and genericity 8. Orbit growth arising from topology 9. Variational aspects of dynamics Part III. Low-Dimensional Phenomena 10. Introduction: What is low dimensional dynamics 11. Homeomorphisms of the circle 12. Circle diffeomorphisms 13. Twist maps 14. Flows on surfaces and related dynamical systems 15. Continuous maps of the interval 16. Smooth maps of the interval Part IV. Hyperbolic Dynamical Systems 17. Survey of examples 18. Topological properties of hyperbolic sets 19. Metric structure of hyperbolic sets 20. Equilibrium states and smooth invariant measures Part V. Sopplement and Appendix 21. Dynamical systems with nonuniformly hyperbolic behavior Anatole Katok and Leonardo Mendoza.

Introduction to the Modern Theory of Dynamical Systems: EQUILIBRIUM STATES AND SMOOTH INVARIANT MEASURES

For a continuous map f on a compact metric space we study the geometry and entropy of the generalized rotation set Rot(Φ). Here Φ = (ϕ1, ..., ϕ
 m
 ) is a m-dimensional continuous potential and Rot(Φ) is the set of all µ-integrals of Φ and µ runs over all f-invariant probability measures. It is easy to see that the rotation set is a compact and convex subset of ℝ
 m
 . We study the question if every compact and convex set is attained as a rotation set of a particular set of potentials within a particular class of dynamical systems. We give a positive answer in the case of subshifts of finite type by constructing for every compact and convex set K in ℝ
 m
 a potential Φ = Φ(K) with Rot(Φ) = K. Next, we study the relation between Rot(Φ) and the set of all statistical limits Rot
 Pt
 (Φ). We show that in general these sets differ but also provide criteria that guarantee Rot(Φ) = Rot
 Pt
 (Φ). Finally, we study the entropy function w ↦ H(w),w ∈ Rot(Φ). We establish a variational principle for the entropy function and show that for certain non-uniformly hyperbolic systems H(w) is determined by the growth rate of those hyperbolic periodic orbits whose Φ-integrals are close to w. We also show that for systems with strong thermodynamic properties (sub-shifts of finite type, hyperbolic systems and expansive homeomorphisms with specification, etc.) the entropy function w ↦ H(w) is real-analytic in the interior of the rotation set.

Geometry and entropy of generalized rotation sets

We establish the existence of ergodic measures of maximal Hausdorff dimension for hyperbolic sets of surface diffeomorphisms. This is a dimension-theoretical version of the existence of ergodic measures of maximal entropy. The crucial difference is that while the entropy map is upper-semicontinuous, the map ν↦ dimHν is neither upper-semicontinuous nor lower-semicontinuous. This forces us to develop a new approach, which is based on the thermodynamic formalism. Remarkably, for a generic diffeomorphism with a hyperbolic set, there exists an ergodic measure of maximal Hausdorff dimension in a particular two-parameter family of equilibrium measures.

https://www.math.wichita.edu/~cwolf/research/preprints/maximal.pdf

Measures of Maximal Dimension for Hyperbolic Diffeomorphisms

Wolf, C., Lafuma, J., Masse, R., Morin, M. and Kellerer, A. M. Neutron RBE for Induction of Tumors with High Lethality in Sprague-Dawley Rats. The effectiveness of fission neutrons is compared to that of γ rays and X rays with regard to the induction of malignancies in male Sprague-Dawley rats. The analysis is based on autopsy results. It is focused on tumors that tend to be present in animals dying early, which is indicative of a high degree of lethality. The relative biological effectiveness (RBE) is deduced from a comparison of the cumulative hazard functions. Different nonparametric models—the constant relative risk model, a time shift model, and an acceleration model—are employed in the comparison, and the resulting values of RBE are seen to be substantially independent of the choice of model. The results are in good agreement with earlier studies of nonlethal lung tumors in the same series of experiments. At neutron doses of 20 to 60 mGy, the RBE of fission neutrons is about 50.

Neutron RBE for Induction of Tumors with High Lethality in Sprague-Dawley Rats

We study the Hausdorff dimension and the pointwise dimension of measures that are not necessarily ergodic. In particular, for conformal expanding maps and hyperbolic diffeomorphisms on surfaces we establish explicit formulas for the pointwise dimension of an arbitrary invariant measure in terms of the local entropy and of the Lyapunov exponents. These formulas are obtained with a direct approach, pertaining to the fundamentals of ergodic theory, which in particular does not require Pesin theory. This allows us to show, for those systems, that the Hausdorff dimension of a (non-ergodic) invariant measure is equal to the essential supremum of the Hausdorff dimensions of the measures in an ergodic decomposition. We also establish corresponding results for non-uniformly hyperbolic diffeomorphisms on surfaces, now using Pesin theory. To the best of our knowledge, the formula for the Hausdorff dimension of an invariant measure along an ergodic decomposition is new even in the case of uniformly hyperbolic dynamics.

Pointwise dimension and ergodic decompositions

Let g be a polynomial automorphism of C2. In a similar way as is done for polynomials inC, we denote byK± the set of points inC2 with bounded forward/ backward orbit under g. We writeJ ± = ∂K± andJ = J+ ∩ J−. We refer toJ ± as the positive/negative Julia set and to J as the Julia set of g. The setJ ± is unbounded, closed, and connected, while J is compact (see [BS2; BS3; FM; HO] for more details). The purpose of the main part of this paper is to show that, under the assumption thatg is a hyperbolic mapping (i.e., the Julia set J is a hyperbolic set for g), the complete information about the Hausdorff dimensions of J+ andJ− is already contained in the Julia set J itself. In particular, the results of Theorem 4.1–4.4 can be summarized by the following result.

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Christian Wolf

Papers

Geometry and entropy of generalized rotation sets

Measures of Maximal Dimension for Hyperbolic Diffeomorphisms

Neutron RBE for Induction of Tumors with High Lethality in Sprague-Dawley Rats

Pointwise dimension and ergodic decompositions

Dimension of Julia sets of polynomial automorphisms of C2.