| _version_ | 1866902096329048064 |
|---|---|
| author | Garycki, Paweł Łukasz |
| author_facet | Garycki, Paweł Łukasz |
| contents | <div> <p>This monograph develops a <strong>systematic framework for perturbing hyperoperation iterators</strong> through rank‑specific modifiers applied to the recursive step, with primary focus on right‑caterpillar constructions.</p> <p>The analysis introduces <strong><span class="math math-inline"><span class="katex"><span class="katex-mathml">kk</span><span class="katex-html"><span class="base"><span class="mord mathnormal">k</span></span></span></span></span>-modified tetration</strong>, defined by the recurrence</p> <div class="math math-display"><span class="katex-display"><span class="katex"><span class="katex-mathml">z[n+1]=b^(z[n]^k), k∈C</span></span></span></div> <p>which unifies standard tetration (<span class="math math-inline"><span class="katex"><span class="katex-html"><span class="base"><span class="mord mathnormal">k</span><span class="mrel">=</span></span><span class="base"><span class="mord">1</span></span></span></span></span>) and <strong>nestrootation</strong> (<span class="math math-inline"><span class="katex"><span class="katex-html"><span class="base"><span class="mord mathnormal">k</span><span class="mrel">=</span></span><span class="base"><span class="mord">−</span><span class="mord">1</span></span></span></span></span>) within a single family. The resulting dynamics are studied via conjugation to the exponential family, yielding an explicit classification of parameter space into attracting basins, period‑doubling cascades, and chaotic regimes.</p> <p>Inverse operators, including the <strong>nest superlog</strong> and <strong>nest superroot</strong>, are defined by swapping functional equations relative to classical tetration inverses. At the tetration (rank‑4) level, the monograph introduces <strong>Pyramidation</strong>, based on balanced bracketing that minimises effective iteration depth. The geometry of such bracketings is formalised using <strong>associahedra</strong>, whose vertices are classified into structural narratives (caterpillars, pyramids, snakes, and concatenations).</p> <p>Analogous perturbations are examined at lower ranks, including nested division at the multiplication level and nested subtraction at the addition level. Analytic continuations to fractional heights are obtained via conjugation, producing complex‑valued oscillatory behaviour even for real bases.</p> <p>The purpose of this deposit is to document the framework and results concerning iterator perturbations as a standalone component of the hyperoperation theory series. No claims are made beyond the scope of the constructions and analyses presented.</p> </div> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_20080012 |
| institution | Zenodo |
| language | eng |
| publishDate | 2026 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | M03 Iterator Perturbations and Nestrootation: A Unied Framework for Modied Hyperoperations Garycki, Paweł Łukasz tetration hyperoperation perturbation <div> <p>This monograph develops a <strong>systematic framework for perturbing hyperoperation iterators</strong> through rank‑specific modifiers applied to the recursive step, with primary focus on right‑caterpillar constructions.</p> <p>The analysis introduces <strong><span class="math math-inline"><span class="katex"><span class="katex-mathml">kk</span><span class="katex-html"><span class="base"><span class="mord mathnormal">k</span></span></span></span></span>-modified tetration</strong>, defined by the recurrence</p> <div class="math math-display"><span class="katex-display"><span class="katex"><span class="katex-mathml">z[n+1]=b^(z[n]^k), k∈C</span></span></span></div> <p>which unifies standard tetration (<span class="math math-inline"><span class="katex"><span class="katex-html"><span class="base"><span class="mord mathnormal">k</span><span class="mrel">=</span></span><span class="base"><span class="mord">1</span></span></span></span></span>) and <strong>nestrootation</strong> (<span class="math math-inline"><span class="katex"><span class="katex-html"><span class="base"><span class="mord mathnormal">k</span><span class="mrel">=</span></span><span class="base"><span class="mord">−</span><span class="mord">1</span></span></span></span></span>) within a single family. The resulting dynamics are studied via conjugation to the exponential family, yielding an explicit classification of parameter space into attracting basins, period‑doubling cascades, and chaotic regimes.</p> <p>Inverse operators, including the <strong>nest superlog</strong> and <strong>nest superroot</strong>, are defined by swapping functional equations relative to classical tetration inverses. At the tetration (rank‑4) level, the monograph introduces <strong>Pyramidation</strong>, based on balanced bracketing that minimises effective iteration depth. The geometry of such bracketings is formalised using <strong>associahedra</strong>, whose vertices are classified into structural narratives (caterpillars, pyramids, snakes, and concatenations).</p> <p>Analogous perturbations are examined at lower ranks, including nested division at the multiplication level and nested subtraction at the addition level. Analytic continuations to fractional heights are obtained via conjugation, producing complex‑valued oscillatory behaviour even for real bases.</p> <p>The purpose of this deposit is to document the framework and results concerning iterator perturbations as a standalone component of the hyperoperation theory series. No claims are made beyond the scope of the constructions and analyses presented.</p> </div> |
| title | M03 Iterator Perturbations and Nestrootation: A Unied Framework for Modied Hyperoperations |
| topic | tetration hyperoperation perturbation |
| url | https://doi.org/10.5281/zenodo.20080012 |