SaddleDrop 0.5.3 by Karl Papadantonakis
http://www.math.cornell.edu/~dynamics/SD
accompanies "Exploring the Parameter Space for Henon Mappings"
paper by John Hubbard and Karl Papadantonakis.
I. CREATING a parameter space for the Henon map
select menu FILE..New Parameter space (command-N).
Window "Create Parameter Plane" appears.
Plane Type: Select which parameter will be varied. There are two parameters (a,c). The Henon map is of the form (x,y) -> (x^2+x-ay,x)
Variable Parameters: Specify ranges for the components of this/these parameter(s)
Fixed Parameters: Specify values for the other parameters, which are constant.
Image Size: Specify size of bitmap. Memory requirement per window is
16 * (horizontal pixels) * (vertical pixels).
Select "true aspect ratio" to constrain pixel size so that each pixel has matching width and height. This is desirable if the parameter plane is complex.
Displayed critical point values: to approximate the connectedness locus, color the fastest-escaping critical point. To approximate the (complex) horseshoes locus, color the slowest-escaping critical point. This uses the following results from the paper:
DEFINITION: the attracting fixed points of Newton's method for critical points of G+ give continuations of the critical points inside K+. In this program we will refer to all such points simply as "critical points".
THEOREM: K is disconnected iff G+ has a critical point on the unstable manifold. I.e. iff the fastest-escaping critical point DOES escape.
CONJECTURE: K has interior iff some critical point is in K+. I.e. for a generic parameter, K is a complex horseshoe iff the slowest-escaping critical point does NOT escape.
Click "Make me one" when all of the above is correctly specified. A new parameter window will appear.
II. IMPROVING the parameter space picture in the window
Initially, the parameter window is blank because no critical points have been given. To look for a critical point, click somewhere in the window. A dynamical plane window appears. See section III below on what to do in the dynamical plane window.
To zoom in the parameter space: press the shift key, and hold while dragging a box.
Options in the ParamPlot menu:
-Remake: make a creation window for the selected parameter window
-CP Animation: display "CP Animation" Window. This must be on the main screen, and the screen must be in thousands of colors for this to work. When a parameter window is selected, the CP Animation window displays the unstable manifold for the parameter at the cursor hotspot, and is automatically refreshed. The picture is zoomed about whichever critical point was used to color the parameter pixel. The color in the center of the deepest blowup should be the same as the color at the cursor.
-Fastest Escaping/Slowest Escaping/Tricolor: choose whether to display escape value of fastest or slowest escaping critical point, or to summarize both (tricolor: yellow=all escape, red=some escape, black=none escape). If you choose Fastest when you were in Slowest, or vice versa, only the most recent critical points subsequently recolored are saved.
-Fake Zoomed: after zooming the parameter space window, the pixel map from the original window is scaled and cropped. This is displayed for reference purposes only. Use this menu option to show/hide this "fake zoomed" image.
Options in the Curves menu:
-Double Fixed point: Show the parameter curve which is the locus where the Henon map has one fixed point, of multiplicity 2.
-Equal Eigenvalue Modulus: Show the parameter curve which is the locus where the maximum eigenvalues of different fixed points have the same absolute value (modulus). This is useful because the program selects (as a matter of convention) whichever eigenvalue has the larger absolute value, and so these curves must be excluded. Fortunately they are degenerate for constant a planes.
-Sinks: Show the parameter curve which is the locus where some fixed point has an eigenvalue of modulus one. Inside some branches of this curve the corresponding fixed point is attracting, so it bounds the period 1 sinks.
III. DYNAMICAL PLANE window
This window displays the unstable manifold for the parameters selected by clicking in a parameter plane window. K+ is colored black, and the complement is colored according to level curves of G+.
By default, the zoom icon (4th from the left) is selected. drag a box to zoom in. If you find a critical point (appearing as a "saddle", "mountain pass", or "space between islands"), first select the Spawn icon (5th from the left). Then click on the critical point. This should start a spreading oil drop in the parent parameter window.
To further improve the picture, optionally select a new parameter, and then look for a different critical point.
Options in the View Menu:
-Open in FractalAsm: Displays the same section of manifold in FractalAsm. If this feature has not been used before, you must select the FractalAsm application (preferably version 0.6.3 or better)
-Re-follow critical point: If this dynamical plane was obtained from a parameter plane by clicking on a parameter point that was reached by some spreading oil drop pass, (either directly or from a zoom of such an image) the critical point which was used to color the clicked parameter is refollowed. This is a quick way
-Select Parameter Window: If this dynamical plane was obtained from a parameter window, that window is selected (brought to the front).
IV. General settings
To change the maximum number of iterations, the maximum number of parameter steps per pixel ("Max subdivisions") or the default dynamical plane window size, choose "General SettingsŠ" from the Edit menu.
V. MONODROMIES
a. definition
For any fixed a , the Hénon mapping is a standard horseshoe map for c sufficiently negative. Define the horseshoe locus to be the component of the set of parameters where the Hénon mapping is hyperbolic containing these real horseshoes.
For H a real horseshoe and p in J, the bi-infinite sequence of 0's and 1's, whose ith entry is 0 if the y-coordinate of H^i(p) is positive and 1 if it is negative is called the symbol of p. The map which associates to each point of J is symbol is an homeomorphism of J onto the shift on two symbols, which conjugates H to the shift.
Given any loop in the horseshoe locus, based at a real horseshoe, following J around the loop induces an automorphism of the 2-shift; points of J might be interchanged. The object of this feature is to compute this automorphism, known as the monodromy of the loop.
b. finite labelling of critical points
Computing with points of J is difficult because they are arbitrarily close to other points of J. Instead we follow the points "in between" points of J, namely the same "critical points" which were used to make the parameter space picture in the first place.
For any real horseshoe there is a "fundamental" critical point which is almost on the x-axis. By convention this critical point gets the name "0." . There are assumed to be infinitely many leading 0's. Trailing digits are left blank; every tail corresponds to a particular point of J under the "umbrella" of this critical point. Similarly "0.0" and "0.1" are names of critical points which land almost on the x-axis after 1 step, and have positive and negative y-coordinates, respectively. Similarly 0.00,0.01,0.11, and 0.10 land almost on the x-axis after 2 steps, etc.
c. using the feature
1. If desired, choose "General Settings" from the "Edit" menu, and set "Parameter Subdiv" (default 50 /100, meaning one step every two pixels) and "Max CP depth" (default 8, meaning all labels 0.,0.0, Š 0.10000000 for a total of 2^(8+1)-1 = 511)
2. Start with a complex a-plane or complex-c plane containing some portion of the real horseshoe locus. (The example "Monodromy" is an a-plane with c=-3 and Re(a)=-0.5 Š -0.3). Using "slowest escaping" mode, follow a few critical points, to get a picture of the horseshoe locus.
3. With the parameter window selected, choose "Monodromy" from the "Paramplot" menu. The option remains selected until a different window is selected, or until a Monodromy is successfully completed.
4. Draw a loop starting at a real horseshoe (where the Julia set contains no points off the real axis) and contained entirely in the horseshoe locus (don't hit any of those reefs in the parameter space. There might be hidden reefs that you don't see). Try to enclose some interesting feature in the loop--if you don't enclose anything you'll just get the identity, boring.
5. A new window titled "permutation" should appear. The diagram in this window represents the permutations of all finite labels up to length 9 (or the value set in General Settings, whichever is less). Each box is the cartesian product of a row interval and a column interval. The row interval represents the umbrella of the starting critical point (with "0." being the entire height of the window), and the column interval is its image. If the umbrella of "0." does not map back into itself, the window will not be square, in order to show those critical points which were mapped to the umbrella of "1.", even though only points which started in the umbrella of "0." were followed.
6. With the permutation window selected, choose "Dump to log" from the "Permutation" menu. This prints a table showing each critical point's label, as well as the label of its image under monodromy automorphism.
d. possible error messages
1. Newton's method did not converge: you might have hit a reef. Check the point where it stopped for non-horseshoeness
2. Itinerary address did not match lexical address: Either you didn't start at a real horseshoe, or at least it wasn't real horseshoe enough. Make c more negative.
VI. Known bugs. To be fixed, eventually, yeah.
1. Critical points are only stored to single precision, so the Refollow command does not always work.
2. Double fixed point curve is not implemented (gracefully) for Re(a)/Re(c) plane except when both imaginary parts are 0.
3. Reboot might be necessary when this program runs out of memory, so give it plenty, and keep an eye on that, and close windows often.
4. Copy dynamical plane not completely implemented. Also dynamical plane has limb-hacking issue that occured in FractalAsm versions 0.4 and earlier. Just use the "Open in FractalAsm" command to get better dynamical plane stuff (e.g. eliminate those bugs, and get smooth coloring and angles).
(other) Bugs? Comments? Questions? E-mail kp30@cornell.edu