At the moment nobody knows the answer – neither for

S₁ = { x ↦ x+1 , x ↦ x³ } (with the two inverse maps added so that the graph is 4–regular)

nor for

S₂ = { x ↦ x+1 , x ↦ 3x , x ↦ x³ } (again with the inverses added).

In other words, it is completely open whether the family of graphs G(p,S₁) (resp. G(p,S₂)) obtained from the prime fields Fₚ (with p ≡ 2 (mod 3) so that x ↦ x³ is a permutation) forms an expander family.

Why we do not know

1. Known proofs of expansion of graphs whose vertices are the elements of Fₚ all come from a linear-group‐picture. The classical example
   {x↦x+1 , x↦−1/x}
   is just the action of the two matrices

      ( 1 1 ) ,  ( 0 −1 )
      ( 0 1 )   ( 1  0 )
   

   on the projective line, and one invokes (Selberg-type) property (τ) for PSL₂(ℤ).
   The same strategy works for any finite collection of rational maps coming from matrices in a fixed linear algebraic group Γ which has property (τ) with respect to the congruence quotients Γ (mod p).
   As soon as one step outside PGL₂ – for instance to the map x↦x³ – there is no such linear interpretation, so the available machinery (property (τ), sum-over-representations, etc.) simply does not apply.

2. The “replacement’’ machinery from additive combinatorics (sum–product estimates, Balog–Szemerédi–Gowers, etc.) does give some mixing information for the walks driven by {x↦x+1,x↦x³}. Typically one can show that after C log p steps the distribution of the walk has L²–mass ≪ p^{−c} on any proper subfield and is already ≪ p^{−c} away from uniform in L∞ or L². Unfortunately the constants c that one gets deteriorate with p, so this gives no uniform spectral gap.

3. Nothing in the structure of the groups generated by the maps helps. For S₂, for instance, the group ⟨ x↦x+1, x↦3x, x↦x³ ⟩ ⊂ Sym(Fₚ) is already gigantic (for many primes it is the full Aₚ), but size alone is not enough for a spectral gap.

What would happen if a negative answer were found?

Assume someone exhibits a sequence of primes pᵢ together with 0-mean, 1-variance functions f_{pᵢ}:F_{pᵢ}→ℂ such that

 ∥ P_{S₁} f_{pᵢ} – f_{pᵢ} ∥₂  –→  0,

i.e. the second largest eigenvalue tends to 1. This would produce large subsets of Fₚ that are simultaneously almost invariant under translation and under cubing. No construction of this kind is known, and it would clash with the prevailing intuition coming from the (sum,product) phenomenon – in particular with the conjectural “sum–product expansion’’ saying that for any A ⊆ Fₚ with p^{ε} < |A| < p^{1–ε} at least one of A+A or A·A must be very large. Hence most experts expect the opposite outcome: there should be a uniform spectral gap, but we currently do not know how to prove it.

To summarise

• We have no genuinely non-linear examples of expander families coming from a fixed finite set of maps on the prime fields.

• The two concrete sets of maps you ask about constitute attractive test cases; they are wide open.

• A positive solution would be a major breakthrough (new techniques beyond property (τ)); a negative one would overturn several currently–held beliefs in additive combinatorics.

References for the current partial results and background
– J. Bourgain, A. Gamburd, P. Sarnak, “Affine linear sieve, expanders, and sum–product”
– J. Bourgain, “More on the sum–product phenomenon in prime fields”
– H. Helfgott, “Growth in groups and graphs tied to group actions”
(These papers give various sub-uniform mixing bounds but no spectral gap for the maps x↦x+1,x↦x³.)