Generating non-jumps from a known one

Let r≥2 be an integer.The real numberα∈[0,1)is a jump for r if there exists a constant c>0 such that for any?>0 and any integer m≥r,there exists an integer n_0(ε,m)satisfying any r-uniform graph with n≥n_0(ε,m)vertices and density at leastα+?contains a subgraph with m vertices and density at leastα+c.A result of Erd?s and Simonovits(1966)and Erdos and Stone(1946)implies that everyα∈[0,1)is a jump for r=2.Erdos(1964)asked whether the same is true for r≥3.Frankl and Rodl(1984)gave a negative answer by showing that1-1/(ε^(r-1))is not a jump for r if r 3 and l>2r.After that,more non-jumps are found by using a method of Frankl and R?dl(1984).Motivated by an idea of Liu and Pikhurko(2023),in this paper,we show a method to construct maps f:[0,1)→[0,1)that preserve non-jumps,i.e.,ifαis a non-jump for r given by the method of Frankl and Rodl(1984),then f(α)is also a non-jump for r.We use these maps to study hypergraph Turán densities and answer a question posed by Grosu(2016).