Generating non-jumps from a known one

Let r ⩾ 2 be an integer. The real number α ∈ [0, 1) is a jump for r if there exists a constant c > 0 such that for any ϵ > 0 and any integer m ⩾ r, there exists an integer n₀(ϵ, m) satisfying any r-uniform graph with n ⩾ n₀ (ϵ, m) vertices and density at least α + ϵ contains a subgraph with m vertices and density at least α + c. A result of Erdős and Simonovits (1966) and Erdős and Stone (1946) implies that every α ∈ [0, 1) is a jump for r = 2. Erdős (1964) asked whether the same is true for r ⩾ 3. Frankl and Rödl (1984) gave a negative answer by showing that \(1-{1 \over {\ell^{r-1}}}\) $$1-\frac{1}{{\ell }^{r-1}}$$ 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 Rödl (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., if α is a non-jump for r given by the method of Frankl and Rödl (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).