Shadowing and Stability of Non-Invertible $p$-adic Dynamics

1. Problem Statement and Context

The paper addresses a gap in the stability theory of dynamical systems, specifically focusing on zero-dimensional spaces (such as the $p$-adic integers $\mathbb{Z}_p$ and $p$-adic numbers $\mathbb{Q}_p$) and non-invertible (non-homeomorphic) maps.

• Background: In positive-dimensional compact metric spaces, the relationship between topological stability and the shadowing property is well-established (Walters' Theorem). For Cantor spaces (zero-dimensional), this relationship was only recently proven for homeomorphisms by Kawaguchi (2019).
• The Gap: Classic results often assume invertibility (homeomorphisms). However, many natural $p$-adic dynamical systems are non-invertible (e.g., shift maps, locally scaling maps). The paper investigates under what conditions these non-invertible $p$-adic maps exhibit the shadowing property (pseudo-orbit tracing) and various forms of structural stability (Lipschitz and topological).

2. Methodology

The authors employ a combination of functional analysis, ultrametric geometry, and fixed-point theory tailored to the specific properties of $p$-adic fields.

• Space Definitions: The study focuses on $X_p \in \{\mathbb{Z}_p, \mathbb{Q}_p\}$ equipped with the $p$-adic norm $\|\cdot\|_p$.
• Key Concepts:
  • Shadowing: A map $f$ has the shadowing property if every $\delta$-pseudo-orbit (a sequence where $f(x_n) \approx x_{n+1}$) is $\epsilon$-close to a true orbit.
  • Stability: Structural stability involves topological conjugacy under small perturbations. The paper distinguishes between topological stability and Lipschitz structural stability (perturbations are restricted to Lipschitz maps).
• Analytical Tools:
  • Banach Fixed-Point Theorem: Used extensively to construct conjugacies and shadowing orbits by defining contraction mappings on sequence spaces ($\ell^\infty$).
  • Right/Left Inverses: The core methodology relies on analyzing maps that admit families of contractions as right or left inverses.
  • Bi-Lipschitz Conditions: The authors investigate the necessity of bi-Lipschitz properties for stability in contractions.
  • Counterexample Construction: To prove optimality, the authors construct specific non-shadowing maps using subshifts of finite type and nowhere locally constant maps.

3. Key Contributions and Results

The paper presents four main theorems that characterize the stability and shadowing properties of non-invertible $p$-adic dynamics.

A. Right-Invertible Maps (Theorem 3.1)

The authors establish sufficient conditions for a continuous map $f: X_p \to X_p$ to be shadowing and stable.

• Condition: $f$ admits a family of right inverses $\{R_i\}_{i \in I}$ which are contractions ($\text{Lip}(R_i) < 1$).
• Covering Condition: The images $R_i(X_p)$ must partition the space $X_p$ (i.e., they are open and pairwise disjoint, and their union is $X_p$).
• Result: Under these conditions, $f$ possesses the shadowing property, is strongly Lipschitz structurally stable, and is topologically stable.
• Applications: This generalizes known results for the shift map $\sigma$ on $\mathbb{Z}_p$ and $(p^{-k}, p^k)$-locally scaling maps. It also applies to homeomorphic dilations on $\mathbb{Q}_p$.

B. Optimality and Counterexamples (Theorem 3.3)

The authors demonstrate that the conditions in Theorem 3.1 are sharp.

• Result: There exists a continuous map $f$ that admits a contraction as a right inverse but fails to be shadowing, Lipschitz structurally stable, or topologically stable.
• Mechanism: The failure occurs when the "covering" condition (partitioning the space) is not met or when the map is not "nowhere locally constant."
• Extension of Kawaguchi: The paper extends Kawaguchi's result to show that any topologically stable map that is nowhere locally constant (no open ball where the map is constant) must have the shadowing property.

C. Left-Invertible Contractions (Theorem 3.6)

The paper investigates contractions $R: X_p \to X_p$ that are left-invertible (injective).

• Condition: $R$ must be bi-Lipschitz (satisfying $c_1 \|x-y\| \le \|R(x)-R(y)\| \le c_2 \|x-y\|$) and have an open image.
• Scaling Property: If $R$ is a "scaling" contraction (preserving the ratio of distances in a specific $p$-adic sense), the stability holds.
• Result:
  • On $\mathbb{Z}_p$: Every open bi-Lipschitz contraction is strongly Lipschitz structurally stable.
  • On $\mathbb{Q}_p$: Every bi-Lipschitz scaling contraction is strongly Lipschitz structurally stable.
• Necessity: The paper provides examples showing that if a contraction is not bi-Lipschitz or does not have an open image, it may fail to be Lipschitz structurally stable, even if it is left-invertible and scaling.

D. Technical Lemmas on Perturbations

A significant portion of the methodology involves proving that these properties are robust under small Lipschitz perturbations ($\phi$).

• Lemma 4.1: If $f$ has a contraction right inverse, then $f + \phi$ (for small $\phi$) also admits a contraction right inverse with similar properties.
• Lemma 4.6 & 4.7: Bi-Lipschitz and scaling properties of contractions are preserved under small Lipschitz perturbations, provided the image remains open.

4. Significance and Impact

1. Extension of Walters' Theorem: The paper successfully adapts the classical link between shadowing and stability to the non-invertible, zero-dimensional $p$-adic setting, a field previously dominated by invertible dynamics.
2. New Classes of Stable Systems: It identifies broad new classes of stable $p$-adic maps, specifically those that are right-invertible via contractions (generalizing shift maps) and specific bi-Lipschitz contractions.
3. Clarification of Conditions: By providing counterexamples (Theorem 3.3 and Example 3.7), the paper clarifies that mere invertibility or the existence of a contraction inverse is insufficient; the topological structure of the image (openness, partitioning) and metric properties (bi-Lipschitz, scaling) are critical for stability.
4. Applications: Since $p$-adic dynamics are used in physics, cryptography, and cognitive science, these stability results provide a theoretical foundation for the robustness of models using these systems against noise or perturbations.

Summary Conclusion

This paper establishes a rigorous framework for the stability of non-invertible $p$-adic dynamical systems. It proves that shadowing and structural stability are guaranteed for maps that can be "decomposed" into families of contracting right inverses that partition the space, and for bi-Lipschitz contractions with open images. The work bridges the gap between classical positive-dimensional stability theory and the emerging field of zero-dimensional $p$-adic dynamics.