Attractors, Vortices and Captives

 Regarding my previous post, I'm inclining toward the following nomenclature:

• an attractor to describe a number that is invariant under the odd-even recursive process
  
  
• a vortex to describe an loop involving two or more numbers under the process
  
  
• a vortical to describe a number that forms part of a vortex
  
  
• a captive to describe a number that eventually leads to an attractor or a vortex
  
  
• N-captive to describe a number that is captive to a number N that is either an attractor or the smallest member of a vortex

The trajectory of 710 leads directly to the attractor 718 so i710 can be described as a 718-captive. The trajectory of 719 leads to 719, 736, 740, 743, 749, 761, 763, 767, 775, 794, 806, 792, 806 and is thus captured by the 792-806 vortex. As a convention, I'll choose the smallest vortical to identify the vortex (although any vortical would do) and so 719 becomes a 792-captive. 

It's clear from the leading number whether the number to captive to an attractor or a vortex but a subscript could be optionally added for numbers with a large number of digits. Thus we could write 718\( _a \)-captive and 782\( _v \)-captive.

Attractors can be visualised as having a solid central core (the attractor itself) with various spokes corresponding to the captives attached to it. For example, 718 is an attractor with four captives: 710, 712, 714 and 716. Each of these four captives is only one step removed from the attractor. This can be represented as shown in Figure 1.

Figure 1: 718 is an attractor with four captives

Some attractors have no captives. For example, in the range from 690 to 889, there are only six attractors: 718, 781, 817, 835, 853 and 871. Of these, only 718 has any captives. On the other hand, some attractors like 87980 with 881 captives can be termed great attractors.

Meanwhile, in the aforementioned range from 690 to 889, every number (apart from the six attractors and four captives) is captive to the small but powerful vortex: 792-806. These two numbers are vorticals and together make up the vortex. Such a vortex could be represented as shown in Figure 2.

Figure 2: the vortex 792-806

A vortex can be shown with its captives attached, although there are too many to show in the case of 792-806. All captives shown can be described as 792-captives. See Figure 3.

Figure 3: vortex, vorticals and captives

Attractors in general could be represented by A and would be equivalent to the number itself and could be differentiated by their subscripts \(A_1, A_2, \dots \). Thus we could write \(A_1=\left \{718 \right \} \). An attractor can also be associated with the set of its captives. For example, an attractor \(A_1\) with \(n\) captives could be associated with the set \(C_1\) such that:$$C_1=\left \{c_1, c_1, \dots , c_{n-1}, c_n \right \}$$ A particular example is \(C_1= \left \{710, 712, 714, 716 \right \} \) where \(A_1= \left \{ 718 \right \} \).

A vortex could be represented by V and would be equivalent to the set of its vorticals v. Thus for a particular vortex \(V_1\), with \(n\) vorticals, could be written as:$$V_1=\left \{ v_1, v_2, \dots , v_{n-1},v_n \right \}$$A particular example would be \(V_1=\left \{792, 806 \right \} \).

A vortex is also associated with the set of its captives and so a particular vortex \(V_1\) with \(n\) captives could be associated with the set \(C_2\) such that:$$C_2=\left \{c_1, c_1, \dots , c_{n-1}, c_n \right \}$$That's about it for this post. I just wanted to establish a consistent and readily understandable notational system. I'm just developing this as I go so there may well be future modifications.