We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.
We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.
Published online by Cambridge University Press: 25 January 2010
The transformation and its properties
We return to our discussion of dynamical systems, and consider an example of great illustrative value for the applications of chaos theory to statistical mechanics, the baker's transformation. For this example, we take the phase-space to be a unit square in the (x,y)-plane, with 0 ≤ x,y≤ 1. The measure-preserving transformation will be an expansion in the x-direction by a factor of 2 and a contraction in the y-direction by a factor of 1/2, arranged in such a way that the unit square is mapped onto itself at each unit of time.
The transformation consists of two steps (see Fig. 7.1): First, the unit square is contracted in the y-direction and stretched in the y-direction by a factor of 2. This doesn't change the volume of any initial region. The unit square becomes a rectangle occupying the region 0≤x ≤ 2; 0 ≤ y ≤ 1/2. Next, the rectangle is cut in the middle and the right half is put on top of the left half to recover a square. This doesn't change volume either. This transformation is reversible except on the lines where the area was cut in two and glued back.
To save this book to your Kindle, first ensure [email protected] is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.
Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.
Find out more about the Kindle Personal Document Service.
To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Dropbox.
To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.
