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April 30, 20260 citationsOpen Access

Analysis of Structural Decay Dynamics and Ergodicity in Collatz Orbits for the Mersenne Class

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ECEmanuel Silva Catarina

Key Points

  • This research aims to analyze the dynamics of Collatz orbits within the context of the Mersenne class.
  • Proving ergodic properties of Collatz orbit M_44497.
  • Calculating metrics over 214,150 steps.
  • Evaluating total variation distance and Lyapunov drift.
  • Using Foster-Lyapunov criteria for recurrence properties.
  • Demonstrated ergodic Markov chain on Z_2 with stationary measure.
  • Determined total variation distance d_TV = 0.031.
  • Identified Lyapunov drift E[Delta lambda] = -0.882.
  • Confirmed finite stopping time N = 214,150.

Abstract

We prove that the Collatz orbit of M₄4497 = 2⁴4497-1 induces an ergodic Markov chain on Z₂ with stationary measure mu (k) = 2^-k. Key metrics over 214, 150 steps: Total Variation distance dTV = 0. 031, Lyapunov drift EDelta lambda = -0. 882, Ev₂ (n+1) = 2. 000. This satisfies Foster-Lyapunov criteria for positive recurrence and proves finite stopping time N = 214, 150. Full code included.

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Cite This Study

Emanuel Silva Catarina (2026) studied this question.

synapsesocial.com/papers/69f2a49d8c0f03fd677639b5https://doi.org/10.5281/zenodo.19854504
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