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Let f_1 and f_2 be affine maps of the N-th dimensional affine space over thecomplex numbers, i.e., f_i(x):=A_i x + y_i (where each A_i is an N-by-N matrixand y_i is a given vector), and let x_1 and x_2 be vectors such that x_i is notpreperiodic under the action of f_i for i=1,2. If none of the eigenvalues ofthe 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 afinite 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 allnon-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 thecomplex numbers (where \Phi_{i,0} is an endomorphism of X and y_i is a givenpoint of X), if none of the eigenvalues of the induced linear actionD\Phi_{i,0} on the tangent space at the identity 0 of X is a root of unity (fori=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 isvarying among all non-negative integers. We give examples to show that theabove condition on eigenvalues is necessary and introduce certain geometricproperties that imply such a condition. Our method involves an analysis ofcertain systems of polynomial-exponential equations and the p-adic exponentialmap for semiabelian varieties.

The orbit intersection problem for linear spaces and semiabelian varieties