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PERIODIC POINTS, LINEARIZING MAPS, AND THE DYNAMICAL MORDELL-LANG PROBLEM
"... Abstract. We prove a dynamical version of the Mordell-Lang conjecture for subvarieties of quasiprojective varieties X, endowed with the action of a morphism Φ: X − → X. We use an analytic method based on the technique of Skolem, Mahler, and Lech, along with results of Herman and Yoccoz from nonarchi ..."
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Abstract. We prove a dynamical version of the Mordell-Lang conjecture for subvarieties of quasiprojective varieties X, endowed with the action of a morphism Φ: X − → X. We use an analytic method based on the technique of Skolem, Mahler, and Lech, along with results of Herman and Yoccoz from nonarchimedean dynamics. 1.
LINEAR RELATIONS BETWEEN POLYNOMIAL ORBITS
, 2008
"... We study the orbits of a polynomial f ∈ C[X], namely the sets {α, f(α), f(f(α)),...} with α ∈ C. We prove that if two nonlinear complex polynomials f, g have orbits with infinite intersection, then f and g have a common iterate. More generally, we describe the intersection of any line in C d with a ..."
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Cited by 9 (4 self)
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We study the orbits of a polynomial f ∈ C[X], namely the sets {α, f(α), f(f(α)),...} with α ∈ C. We prove that if two nonlinear complex polynomials f, g have orbits with infinite intersection, then f and g have a common iterate. More generally, we describe the intersection of any line in C d with a d-tuple of orbits of nonlinear polynomials, and we formulate a question which generalizes both this result and the Mordell–Lang conjecture.
PERIODS OF RATIONAL MAPS MODULO PRIMES
, 2011
"... Let K be a number field, let ϕ ∈ K(t) be a rational map of degree at least 2, and let α, β ∈ K. We show that if α is not in the forward orbit of β, then there is a positive proportion of primes p of K such that α mod p is not in the forward orbit of β mod p. Moreover, we show that a similar result ..."
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Cited by 9 (3 self)
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Let K be a number field, let ϕ ∈ K(t) be a rational map of degree at least 2, and let α, β ∈ K. We show that if α is not in the forward orbit of β, then there is a positive proportion of primes p of K such that α mod p is not in the forward orbit of β mod p. Moreover, we show that a similar result holds for several maps and several points. We also present heuristic and numerical evidence that a higher dimensional analog of this result is unlikely to be true if we replace α by a hypersurface, such as the ramification locus of a morphism ϕ: P n → P n.
A case of the dynamical Mordell–Lang conjecture
- MATHEMATISCHE ANNALEN
, 2012
"... We prove a special case of a dynamical analogue of the classical Mordell–Lang conjecture. Specifically, let ϕ be a rational function with no periodic critical points other than those that are totally invariant, and consider the diagonal action of ϕ on (P 1) g. If thecoefficientsofϕ are algebraic, we ..."
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Cited by 9 (1 self)
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We prove a special case of a dynamical analogue of the classical Mordell–Lang conjecture. Specifically, let ϕ be a rational function with no periodic critical points other than those that are totally invariant, and consider the diagonal action of ϕ on (P 1) g. If thecoefficientsofϕ are algebraic, we show that the orbit of a point outside the union of the proper preperiodic subvarieties of (P 1) g has only finite intersection with any curve contained in (P 1) g. We also show that our result holds for indecomposable polynomials ϕ with coefficients in C. Our proof uses results from p-adic dynamics together with an integrality argument. The extension to polynomials defined over C uses the method of specialization coupled with some new results of Medvedev and Scanlon for describing the periodic plane curves under the action of (ϕ, ϕ) on A².
Dynamical degree, arithmetic entropy, and canonical heights for dominant rational self maps of projective space
- ERGODIC THEORY AND DYN. SYS. 34 (2014) 647–678
, 2014
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ON RITT’S POLYNOMIAL DECOMPOSITION THEOREMS
, 2008
"... Ritt studied the functional decomposition of a univariate complex polynomial f into prime (indecomposable) polynomials, f = u1 ◦ u2 ◦ · · · ◦ ur. His main achievement was a procedure for obtaining any decomposition of f from any other by repeatedly applying certain transformations. However, Ritt ..."
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Ritt studied the functional decomposition of a univariate complex polynomial f into prime (indecomposable) polynomials, f = u1 ◦ u2 ◦ · · · ◦ ur. His main achievement was a procedure for obtaining any decomposition of f from any other by repeatedly applying certain transformations. However, Ritt’s results provide no control on the number of times one must apply the basic transformations, which makes his procedure unsuitable for many theoretical and algorithmic applications. We solve this problem by giving a new description of the collection of all decompositions of a polynomial. One consequence is as follows: if f has degree n> 1 but f is not conjugate by a linear polynomial to either X n or ±Tn (with Tn the Chebychev polynomial), and if the composition a ◦ b of polynomials a, b is the k th iterate of f for some k> log 2 (n + 2), then either a = f ◦ c or b = c ◦ f for some polynomial c. This result has been used by Ghioca, Tucker and Zieve to describe the polynomials f, g having orbits with infinite intersection; our results have also been used by Medevedev and Scanlon to describe the affine varieties invariant under a coordinatewise polynomial action. Ritt also proved that the sequence (deg(u1),..., deg(ur)) is uniquely determined by f, up to permutation. We show that in fact, up to permutation, the sequence of permutation groups (G(u1),..., G(ur)) is uniquely determined by f, where G(u) = Gal(u(X)−t, C(t)). This generalizes both Ritt’s invariant and an invariant discovered by Beardon and Ng, which turns out to be equivalent to the subsequence of cyclic groups among the G(ui).
Decompositions of Laurent polynomials
"... Abstract. In the 1920’s, Ritt studied the operation of functional composition g ◦ h(x) = g(h(x)) on complex rational functions. In the case of polynomials, he described all the ways in which a polynomial can have multiple ‘prime factorizations ’ with respect to this operation. Despite significant e ..."
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Abstract. In the 1920’s, Ritt studied the operation of functional composition g ◦ h(x) = g(h(x)) on complex rational functions. In the case of polynomials, he described all the ways in which a polynomial can have multiple ‘prime factorizations ’ with respect to this operation. Despite significant effort by Ritt and others, little progress has been made towards solving the analogous problem for rational functions. In this paper we use results of Avanzi–Zannier and Bilu–Tichy to prove analogues of Ritt’s results for decompositions of Laurent polynomials, i.e., rational functions with denominator x n. 1.
THE MORDELL–LANG QUESTION FOR ENDOMORPHISMS OF SEMIABELIAN VARIETIES
"... Abstract. The Mordell–Lang conjecture describes the intersection of a finitely generated subgroup with a closed subvariety of a semiabelian variety. Equivalently, this conjecture describes the intersection of closed subvarieties with the set of images of the origin under a finitely generated semigro ..."
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Abstract. The Mordell–Lang conjecture describes the intersection of a finitely generated subgroup with a closed subvariety of a semiabelian variety. Equivalently, this conjecture describes the intersection of closed subvarieties with the set of images of the origin under a finitely generated semigroup of translations. We study the analogous question in which the translations are replaced by algebraic group endomorphisms (and the origin is replaced by another point). We show that the conclusion of the Mordell–Lang conjecture remains true in this setting if either (1) the semiabelian variety is simple, (2) the semiabelian variety is A 2, where A is a one-dimensional semiabelian variety, (3) the subvariety is a connected one-dimensional algebraic subgroup, or (4) each endomorphism has diagonalizable Jacobian at the origin. We also give examples showing that the conclusion fails if we make slight modifications to any of these hypotheses. 1.
On maximal decompositions of rational functions
"... Abstract. In this paper we extend the first Ritt theorem about decompositions of polynomials to rational functions the monodromy group of which contains a cyclic subgroup with at most two orbits. Besides, we give a detailed analysis of the simplest examples of rational functions, related to finite s ..."
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Abstract. In this paper we extend the first Ritt theorem about decompositions of polynomials to rational functions the monodromy group of which contains a cyclic subgroup with at most two orbits. Besides, we give a detailed analysis of the simplest examples of rational functions, related to finite subgroups of Aut(CP 1), for which the first Ritt theorem fails to be true. 1.
ON A UNIFORM BOUND FOR THE NUMBER OF EXCEPTIONAL LINEAR SUBVARIETIES IN THE DYNAMICAL MORDELL–LANG CONJECTURE
"... Abstract. Let φ: P n → P n be a morphism of degree d ≥ 2 defined over C. The dynamical Mordell–Lang conjecture says that the intersection of an orbit Oφ(P) and a subvariety X ⊂ P n is usually finite. We consider the number of linear subvarieties L ⊂ P n such that the intersection Oφ(P) ∩ L is “larg ..."
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Abstract. Let φ: P n → P n be a morphism of degree d ≥ 2 defined over C. The dynamical Mordell–Lang conjecture says that the intersection of an orbit Oφ(P) and a subvariety X ⊂ P n is usually finite. We consider the number of linear subvarieties L ⊂ P n such that the intersection Oφ(P) ∩ L is “larger than expected. ” When φ is the d th-power map and the coordinates of P are multiplicatively independent, we prove that there are only finitely many linear subvarieties that are “super-spanned ” by Oφ(P), and further that the number of such subvarieties is bounded by a function of n, independent of the point P and the degree d. More generally, we show that there exists a finite subset S, whose cardinality is bounded in terms of n, such that any n+1 points in Oφ(P)�S are in linear general position in P n. 1. The Dynamical Mordell–Lang Conjecture The classical Mordell conjecture says that a curve C of genus g ≥ 2 defined over a number field K has only finitely many K-rational points. One may view C as embedded in its Jacobian J, and then Mordell’s conjecture may be reformulated as saying that C intersects the finitely generated group J(K) in only finitely many points. Taking this viewpoint, Lang
