![]() ![]() In the next few lemmas, we will show that k X maps the exceptional fibers as shown in Figure 3.1. Real and complex dynamics of a family of birational maps of the plane: The golden mean subshift American Journal of Mathematics. Let k X : X → X denote the induced map on the complex manifold X. Dynamics of a two parameter family of plane birational maps: Maximal entropy Journal of Geometric Analysis. We use homogeneous coordinates by identifying a point ( t, y ) ∈ C 2 with ∈ P 2. ![]() For E 1 and P j, 1 ≤ j ≤ n − 1 we use local coordinate systems defined in (2.2–4). That is, in a neighborhood of Q we use a ( ξ 1, v 1 ) = ( t 2 / y, y / t ) coordinate system. The iterated blow-up of p 1, …, p n − 1 is exactly the process described in §2, so we will use the local coordinate systems defined there. (iv) blow up p j : = E 1 ∩ P j − 1 with exceptional fiber P j for 2 ≤ j ≤ n − 1. (iii) blow up p 1 : = E 1 ∩ C 1 and let P 1 denote the exceptional fiber over p 1, (ii) blow up q : = E 1 ∩ C 4 and let Q denote the exceptional fiber over q, We define a complex manifold π X : X → P 2 by blowing up points e 1, q, p 1, …, p n − 1 in the following order: (i) blow up e 1 = and let E 1 denote the exceptional fiber over e 1, ![]() We comment that the construction of X and ~ k can yield further information about the dynamics of k (see, for instance, and ). The general existence of such a map ~ k when δ ( k ) > 1 was shown in. This method has also been used by Takenawa. By the birational invariance of δ (see and ) we conclude that δ ( k F ) is the spectral radius of ~ k ∗. There is a well defined map ~ k ∗ : P i c ( X ) → P i c ( X ), and the point is to choose X so that the induced map ~ k satisfies ( ~ k ∗ ) n = ( ~ k n ) ∗. If Xis the golden mean Z subshift on f0 1gwhere adjacent 1s are prohibited, then E(000) is the set of all legal con gurations on Znf0 1 2g, which is identi ed with the set of all f0 1gsequences xwhich have no adjacent 1s, with the exception that x 0 x 1 1 is allowed. That is, we find a birational map φ : X → P 2, and we consider the new birational map ~ k = φ ∘ k F ∘ φ − 1. Hermitian -theory of the integers Bloch, Spencer, Helene Esnault, and Marc Levine. The approach we use here is to replace the original domain P 2 by a new manifold X. Real and complex dynamics of a family of birational maps of the plane: The golden mean subshift BERRICK, A. As was noted by Fornæss and Sibony, if there is an exceptional curve whose orbit lands on a point of indeterminacy, then the degree is not multiplicative: ( d e g ( k F ) ) n ≠ d e g ( k n F ). That is, there are exceptional curves, which are mapped to points and there are points of indeterminacy, which are blown up to curves. Acta Mathematica, vol 125, pp 193-225 (2020).We will analyze the family k F by inspecting the blowing-up and blowing-down behavior. (with Eric Bedford) Energy and invariant measure for birational surface maps, Duke Mathematical Journal 128 (2005), pages 338-368. Experimental Math, vol 30, pp 172-190 (2021). (with Eric Bedford) Real and complex dynamics of a family of birational maps of the plane: the golden mean subshift, American Journal of Mathematics 127 (2005), pages 595-646. \Real and complex dynamics of rational surface automorphisms". Favre, Dynamics of bimeromorphic maps of surfaces, Amer. (18) Eric Bedford and Jeffrey Diller, Real and complex dynamics of a family of birational maps of the plane: the golden mean subshift, Amer.(16) Jeffrey Diller, Daniel Jackson, and Andrew Sommese, Invariant curves for birational surface maps, Trans.(8) Jeffrey Diller, Romain Dujardin, and Vincent Guedj, Dynamics of meromorphic mappings with small topological degree II: Energy and invariant measure, Comment.(3) Jeffrey Diller and Jan-Li Lin, Rational surface maps with invariant meromorphic two-forms, Math.For this last purpose, computer experimentation often plays a crucial role in building the understanding and intuition needed to make mathematical progress. I am also interested in understanding the dynamics of particular examples in much greater detail. I am currently interested in extending this picture to higher dimensions and non-invertible rational maps. ![]() So far, I have mostly concentrated on the case of plane birational maps, and this work has led to a general probabilistic picture for the dynamics of such maps. Using tools from pluripotential theory, complex algebraic geometry and dynamical systems, my goal is to understand the behavior of rational maps of two or more variables under iteration. D., University of Michigan, 1993 Research GroupsĪlgebra and Algebraic Geometry, Analysis and Partial Differential Equations Research AreaĬomplex Analysis and Geometry, Dynamical Systems Bio Research Interests ![]()
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