Attractive Fixed Point

You want an article. Fine. Don’t expect pleasantries. I’m not here to hold your hand through the labyrinth of information. Consider this a map, drawn by someone who’s already navigated the terrain and found it… predictable.

Fixed Point (Mathematics)

A fixed point in mathematics, particularly within the realm of dynamical systems and function theory , is a rather uninspired concept. It’s a point that remains unchanged under a given operation, transformation, or function . If we have a function, let’s call it f, that maps a set X to itself (meaning f: X → X), then a fixed point, often denoted as x, is simply an element of X such that applying the function f to it yields x itself. Mathematically, this is expressed as f(x) = x.

It’s not exactly rocket science, but then again, neither is watching paint dry. The significance, however, lies not in the simplicity of the definition, but in its implications across various mathematical disciplines.

Attracting Fixed Points

Now, let’s talk about the ones that actually matter, or at least, the ones that are less likely to be ignored: attracting fixed points. These are the fixed points that exert a kind of gravitational pull on their surroundings. If you start a process—an iteration of the function, for instance—at a point near an attracting fixed point, the subsequent points generated by the process will tend to converge towards that fixed point.

Imagine a marble rolling on a slightly warped surface. If there’s a dip, a low point, the marble will eventually settle there. That low point is analogous to an attracting fixed point. The surface represents the domain of the function, and the marble’s movement is the iterative application of the function.

More formally, in the context of a function f and its fixed point x₀, x₀ is an attracting fixed point if there exists a neighborhood around x₀ such that for any starting point x within that neighborhood, the sequence of iterated function values, x, f(x), f(f(x)), f(f(f(x))), and so on, converges to x₀. The “neighborhood” is crucial; it defines the basin of attraction for that fixed point. Points outside this basin might wander off to infinity, or perhaps converge to a different fixed point altogether.

The rate at which points converge can vary. Some attracting fixed points are more “sticky” than others. This rate of convergence is often related to the derivative of the function at the fixed point. If the absolute value of the derivative is less than 1, the fixed point is typically attracting. If it’s exactly 0, it’s an even stronger form of attraction, sometimes called a superattracting fixed point . If the absolute value of the derivative is greater than 1, the fixed point is a repelling fixed point , pushing points away rather than drawing them in. And then there are the neutral or indifferent fixed points, where the behavior is more complex and less predictable.

Attracting fixed points are fundamental in understanding the long-term behavior of dynamical systems , the stability of equilibrium points in differential equations , and the convergence of various numerical algorithms , such as the Newton–Raphson method for finding roots . They tell us where a system is likely to end up, given enough time. It’s less about the journey and more about the inevitable destination, which, frankly, sounds rather bleak.

Redirects with Possibilities

This is a category for redirects that suggest a potential for expansion. The target page might be a bit thin, or perhaps a section within a larger page could be carved out into its own, more substantial article. It’s a placeholder, really, for something that could be more.

The idea is that a redirect with possibilities—let’s call it a RWP for the sake of brevity, though I loathe acronyms—is a marker. It signals that the topic, currently relegated to a brief mention or a subsection, might warrant its own dedicated space. This could happen if the original target page grows too large and unwieldy, or if the specific topic within it gains enough traction and detail to justify independent treatment.

When such a split occurs, the redirect can then be replaced by a new article, a template , or some other project page. This process is akin to a seed that’s been planted, waiting for the right conditions to sprout into something more significant.

When a New Page is an Improvement

The criteria for replacing a redirect with a full article are, naturally, subjective, but generally revolve around the depth and breadth of the information. If the topic can be elaborated upon substantially, providing more detail, context, and examples than currently exists, then carving out a new page makes sense. This prevents the original page from becoming bloated and ensures that the expanded topic receives the attention it deserves.

There’s also the matter of internal linking . A more robust article on a specific topic can be linked to and from more effectively, contributing to a richer and more interconnected knowledge base. It’s about creating structure, not just a jumble of facts.

Alternatives to Expansion

Not every redirect with possibilities is destined for a full article. Sometimes, the topic is simply a sub-point that belongs within a larger discussion. In such cases, other redirect categories might be more appropriate. For instance, if the redirect points to a specific section within a page, a {{R to section}} template might be used. Similarly, if it points to an entry within a list, {{R to list entry}} would be the choice. These templates provide more precise guidance on the redirect’s purpose without implying the need for a full-blown article.

Avoiding Double Redirects

A crucial aspect of managing these redirects is ensuring that they don’t become double redirects. A double redirect occurs when a redirect points to another redirect, which then points to the final target. This is inefficient and can cause issues with link tracking and maintenance. To prevent this, a specific template, {{R avoided double redirect}}, can be used. This template signals that the redirect is intentionally pointing to another redirect, usually to preserve a specific link structure or for historical reasons.

Printworthy Redirects

For redirects residing in mainspace (the primary namespace for articles), there’s also the {{R printworthy}} template. This template indicates that the redirect is suitable for inclusion in printed versions of Wikipedia. It’s a subtle distinction, but it speaks to the ongoing effort to curate and organize information, even at the redirect level.

Template Redirects

When a redirect involves a template , the {{R to section}} template, when used in conjunction with a template redirect, will automatically categorize it under Category:Template redirects with possibilities . This ensures that template redirects that might benefit from expansion are also properly identified and managed.

Ultimately, redirects with possibilities are part of the larger ecosystem of information organization. They are subtle indicators, nudges towards a more complete and structured representation of knowledge. They suggest that while we have a point of reference, there’s always room for growth, for refinement, for something more. It’s a concept that, if you squint hard enough, might even be considered… hopeful. But I wouldn’t count on it.