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- W2341171666 abstract "Let f_1 and f_2 be affine maps of the N-th dimensional affine space over the complex numbers, i.e., f_i(x):=A_i x + y_i (where each A_i is an N-by-N matrix and y_i is a given vector), and let x_1 and x_2 be vectors such that x_i is not preperiodic under the action of f_i for i=1,2. If none of the eigenvalues of the matrices A_i is a root of unity, then we prove that the set of pairs (n_1,n_2) of non-negative integers such that f_1^{n_1}(x_1)=f_2^{n_2}(x_2) is a finite union of sets of the form (m_1k + ell_1, m_2k + ell_2) where m_1, m_2, ell_1, ell_2 are given non-negative integers, and k is varying among all non-negative integers. Using this result, we prove that for any two self-maps Phi_i(x) := Phi_{i,0}(x)+y_i on a semiabelian variety X defined over the complex numbers (where Phi_{i,0} is an endomorphism of X and y_i is a given point of X), if none of the eigenvalues of the induced linear action DPhi_{i,0} on the tangent space at the identity 0 of X is a root of unity (for i=1,2), then for any two non-preperiodic points x_1,x_2, the set of pairs (n_1,n_2) of non-negative integers such that Phi_1^{n_1}(x_1) = Phi_2^{n_2}(x_2) is a finite union of sets of the form (m_1k + ell_1, m_2k + ell_2) where m_1,m_2,ell_1,ell_2 are given non-negative integers, and k is varying among all non-negative integers. We give examples to show that the above condition on eigenvalues is necessary and introduce certain geometric properties that imply such a condition. Our method involves an analysis of certain systems of polynomial-exponential equations and the p-adic exponential map for semiabelian varieties." @default.
- W2341171666 created "2016-06-24" @default.
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- W2341171666 date "2016-04-09" @default.
- W2341171666 modified "2023-09-27" @default.
- W2341171666 title "The orbit intersection problem for linear spaces and semiabelian varieties" @default.
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- W2341171666 hasPublicationYear "2016" @default.
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