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An evolutionary debate contrasts the importance of genetic convergence versus genetic redundancy. In genetic convergence, the same adaptive trait evolves because of similar genetic changes. In genetic redundancy, the adaptive trait evolves using different genetic combinations, and populations might not share the same genetic changes. Here we address this debate by examining single nucleotide polymorphisms (SNPs) associated with the rapid evolution of character displacement in Anolis carolinensis populations inhabiting replicate islands with and without a competitor species (1Spp and 2Spp islands, respectively). We identify 215-outliers SNPs that have improbably large FST values, low nucleotide variation, greater linkage than expected and that are enriched for genes underlying animal movement. The pattern of SNP divergence between 1Spp and 2Spp populations supports both genetic convergence and genetic redundancy for character displacement. In support of genetic convergence: all 215-outliers SNPs are shared among at least three of the five 2Spp island populations, and 23% of outlier SNPS are shared among all five 2Spp island populations. In contrast, in support of genetic redundancy: many outlier SNPs only have meaningful allele frequency differences between 1Spp and 2Spp islands on a few 2Spp islands. That is, on at least one of the 2Spp islands, 77% of outlier SNPs have allele frequencies more similar to those on 1Spp islands than to those on 2Spp islands. Focusing on genetic convergence is scientifically rigorous because it relies on replication. Yet, this focus distracts from the possibility that there are multiple, redundant genetic solutions that enhance the rate and stability of adaptive change. [ABSTRACT FROM AUTHOR]