2012/10/16 by Ronen Peretz, Peretz, Ronen
Mathematics · Physics and Astronomy · #Advanced Differential Equations and Dynamical Systems #Mathematical Dynamics and Fractals #Nonlinear Waves and Solitons #math.AG
paper · pdf · doi:10.48550/arxiv.1210.4465
arxiv created 2012/10/16 · arxiv updated 2012/10/17
We hope to be able (in the future) to carefully analyze this structure and to tie the Jacobian Conjecture in dimension two to certain Zeta functions, thereby invoking a powerful arithmetic machinery to handle the two dimensional Jacobian Conjecture. Let us denote by \rm et(ℂ2) the semigroup of two dimensional Keller mappings. We would like to prove something like the following: a) That there exists an infinite index set I, and a family of mappings indexed by I, \Fi | i∈ I\ ⊂ \rm et(ℂ2) such that \rm et(ℂ2)=\rm Aut(ℂ2)∪\bigcupi∈ I RFi(\rm et(ℂ2)), where if i≠ j then RFi(\rm et(ℂ2))∩ RFj(\rm et(ℂ2))=∅. b) That the parallel representation to the representation described in (a) above holds true, this time with respect to the left composition operators LGj. These two claims will be the basis for a fractal structure on \rm et(ℂ2) because the pieces RFi(\rm et(ℂ2)) are similar to each other in the sense that they are homeomorphic, and we further have the scaling property of self-similarity, namely RF(\rm et(ℂ2)) is homeomorphic to its proper subspace RG∘ F(\rm et(ℂ2)) that is homeomorphic to its proper subspace RH∘ G∘ F(\rm et(ℂ2)) etc . This is the right place to remark that the purpose of the current paper is to start and develop the parallel theory for entire functions in one complex variable. Results in this setting will hint that there are hopes to accomplish the above objective.