The Full Progression Example for Question 5.6
Abstract
A weakly mixing compact metric system with neighborhood two-shadowing and without dense nonempty small periodic sets.
Jonathan Meddaugh and Elyssa Stephens, Shadowing with small diameter sets of bounded cardinality, arXiv:2509.04312v1, Question 5.6 asks: Is there a weakly mixing system which has the neighborhood N-shadowing property (N>1) that does not have dense small periodic sets? Use the intended nonempty meaning of a small periodic set. The printed definition omits nonemptiness; allowing the empty set would trivialize it. The phase space is X={x in {0,1}^N : for every s>=0,p>=1 some j<=p^2+2 has x(s+jp)=0}, with the genuine first-disagreement metric, and the one-sided left shift sigma. No homeomorphism hypothesis is imposed by the source’s compact metric system definition.
The arithmetic-progression construction, constants and rotation deletion method are attributed to Pro.
Theorem 1.1 (Weak mixing on all nonempty product open sets).
Proof. Machine-checked in Lean as D5/S1/Dynamics/Shadowing/ProgressionExample.progression_shift_product_transitive (✓ std3). ∎
Source. Repository-derived.
Commentary.
For this generic statement ell is any natural-valued length function satisfying ell(p)>=3 for every positive p. Set(T) denotes subsets of T and iterate(f,n,x) denotes f^[n](x). For arbitrary nonempty open U,V in ProgressionPoint(ell) times ProgressionPoint(ell), choose contained nonempty open rectangles. Apply open-set mixing to each coordinate pair. The maximum of their two thresholds supplies one common hitting time and two points, proving transitivity of the square map.
In the endpoint formula, X is ProgressionPoint(quadraticProgressionLength) and sigma is progressionShift(quadraticProgressionLength). Set(X) denotes all subsets of X; iterate(f,n,x) denotes f^[n](x), sigma^p denotes p-fold iteration, and image(f,B) denotes the direct image. MapsTo(f,B,B) means image(f,B) is a subset of B. NeighborhoodTwoShadowing(sigma) is the existing predicate: for every epsilon>0 there is delta>0 such that every infinite natural-time delta-pseudo-orbit admits one same nonempty finite A in X with card(A)<=2 and initial diameter<epsilon, and at every natural time its distance to the shifted set is <epsilon. The minimizing branch may vary with time.
Theorem 1.2 (The full compact-system existence conclusion).
Proof. Machine-checked in Lean as D5/S1/Dynamics/Shadowing/ProgressionExample.quadratic_progression_example (✓ std3). ∎
Resolves. Problems/meddaugh-stephens-2025-mixing-small-periodic-sets (proved) by D5/S1/Dynamics/Shadowing/ProgressionExample.quadratic_progression_example.
Source. Repository-derived.
Commentary.
The all-zero configuration witnesses nonemptiness, and the closed finite-alphabet carrier is compact. Zero-prepending gives surjectivity, while zero-gap joining gives mixing and hence product transitivity. The exact original neighborhood contract holds with the single fixed N=2 for all accuracies and all infinite natural-time pseudo-orbits. Take U=[1], containing 10-infinity. For every p>0 it contains no nonempty B with sigma^p(B) subset B; hence it contains no nonempty compact B with sigma^p(B)=B. One nonempty open obstruction negates density of nonempty small periodic sets. This is the complete intended existence conclusion, not a finite instance or a conditional mask statement.