You can download a tutorial introduction to this horseshoe calculator.

This page contains some utilities for calculating quantities associated with Smale's horseshoe map. The following are available:

**Orbit calculator:** Calculates heights, decorations, rotation intervals, decoration invariants, and a number of other quantities for horseshoe periodic orbits. You can choose which quantities you want to compute, which orbits you want to compute them for (either a list of orbits you provide, or all orbits of given periods), and how the results are to be presented (list, HTML table, or LaTeX table).

**Height calculator:** Calculates the heights of sequences whose initial segments you provide: or, if the initial segment doesn't determine the height, the interval which the height belongs to.

**Decoration calculator:** Calculates the scope of each of a list of decorations, and also the number and/or the codes of all periodic orbits with the given decorations whose height is equal to the scope.

**General notes:** When input of a list of codes, decorations, or initial segments is required in a text area, you can separate different items in the list with either spaces or new lines. Anything following the symbol ! in any line will be commented out.

Please send any questions, or details of bugs, to Toby Hall .

1. Check the boxes below to specify the information you want for each orbit. If you want the information presented in a particular order, put numbers in the text boxes by each item. Items with lower numbers will be presented before items with higher numbers. You can also specify the column headings in table views in the second text boxes (and include $ signs for the LaTeX table if you want). For decoration invariants, use the placeholder ! to indicate where the decoration itself should appear (e.g. you could put `$r^{!}$` ).

Quantities which are braid type invariants are marked with a * below

2. If you checked the *Decoration Invariants* box, then specify the decorations for which you want to calculate the associated invariants in the text area below. Type ***** for the NBT decoration and **e** for the empty decoration. **Note that these are only known to be invariants for the subset of lone decorations. **

3. Select the periodic orbits you want to process. Click the appropriate radio button, and then either enter the codes/heights and decorations of the orbits you want to process in the text area below, or select the checkboxes to process all orbits of the given periods.

Depending on the box checked above, either enter codes below, or enter a collection of heights and decorations (in any order): one orbit of each compatible height and decoration listed will be processed. Type

4. Select the type of output you want, then press the `Process Orbits` button. (List output is only suitable for relatively small numbers of orbits.)

Orbit calculator Height calculator Decoration calculator

Enter one or more initial segments of sequences below and press the `Calculate Height` button. The script will return the height of each sequence if it is determined by the initial segment, otherwise it will return the interval which the height lies in.

Orbit calculator Height calculator Decoration calculator

1. Check the boxes below to specify the information you want for each decoration

2. Select the appropriate radio button, and then either enter the decorations you want to process in the box below, or select the checkboxes to process all decorations of the given lengths. Type ***** for the NBT decoration and **e** for the empty decoration.

3. Select the type of output you want, then press the `Process Decorations` button. (List output is only suitable for relatively small numbers of decorations.)

Orbit calculator Height calculator Decoration calculator