On Sat, Apr 21, 2012 at 6:52 PM, Bill Gosper <billgosper@gmail.com> wrote:
http://gosper.org/x=1, y=1, dmin=emin=3o2, dmax=emax=8o3, dg=eg=1o2574, de<=49o12.png , corrects and refines the "δ-ℇ plot" on p23 of http://www.blurb.com/books/2172660 (also Neil's blog,
http://nbickford.files.wordpress.com/2011/03/x1-y1-dminemin3o2-dmaxemax8o3-d... ). It's a map of the periods of the Minsky circle algorithm, starting with x0=1, y0=1, with multipliers 3/2 ≤ δ,ℇ ≤ 8/3. With a zoomable, nonblurring viewer, I find it fascinating and surprising. The (δ,ℇ) which blow up exponentially (because δ ℇ>4) are shown in white. But note that some colored rectangles transgress δ ℇ = 4 ! The floor function in the Minsky iteration can actually tame an infinite sequence. Contrariwise, imposing the floor operation on x0=1, y0=1/2, δ 3^(n+1), ℇ = 3^-n changes a mere period 3 into infinite linear growth! Any such growers in the graphic appear black, along with any orbits longer than 10000, of which there are >57000 (in this plot).
Prior to this hi-res plot, I had conjectured that no rectangle had two corners transgressing the hyperbola. Julian completed my humiliation by finding the period 159 region {x=1, y=1, 83/49 ≤ δ < 61/36, 85/36 ≤ ℇ < 111/47} with *three* transgressing corners. I still conjecture the impossibility of four. [...]
Talk about hanging by a thread! This is Corey's region finder: In[164]:= ExpandAll[ RiskyCongruenceRegion[1, 2, 38/29 + 1/9999, 58/19 + 1/9999]] Out[164]= {MatrixForm[{(Inequality[55/42, Less, \[Delta], LessEqual, 38/29] && Inequality[-53 + 72/\[Delta], LessEqual, y0, Less, 31 - 38/\[Delta]]) || (Inequality[38/29, Less, \[Delta], LessEqual, 97/74] && Inequality[34 - 42/\[Delta], LessEqual, y0, Less, 31 - 38/\[Delta]]) || (97/74 < \[Delta] < 101/77 && Inequality[34 - 42/\[Delta], LessEqual, y0, Less, -43 + 59/\[Delta]]), (Inequality[58/19, Less, \[Epsilon], LessEqual, 171/56] && Inequality[-18 + 58/\[Epsilon], LessEqual, x0, Less, 20 - 58/\[Epsilon]]) || (171/56 < \[Epsilon] < 113/37 && Inequality[38 - 113/\[Epsilon], LessEqual, x0, Less, -36 + 113/\[Epsilon]])}], 72} In[163]:= Simplify[% /. x0 -> 1 /. y0 -> 2] Out[163]= {MatrixForm[{38/29 < \[Delta] < 59/45, 58/19 < \[Epsilon] < 113/37}], 72} I.e., three corners out and one corner *on* the hyperbola, but it's and open rectangle, and thus entirely out! --rwg PS, a wider view of that "cave" plot shows a remarkable 1D transgression<http://gosper.org/x=1,y=2,1o4led,ele4,dg=eg=1o720.png> at (4/3,3).