The logistic map is used to update F, while the

circle map is employed to generate chaotic generations used during search.

Lessons use exit cards, choice boards, cubing, graphic organizers, learning contracts, and tiers, and include instructions, examples, tips on how and when to use them, reproducible handouts, how they fit with their

CIRCLE MAP model, and comments from the teacher who developed them on how to use strategies and how students responded, as well as comments from other teachers who used them.

Framework Task/Activity Recording / Component Thinking Tool Tuning In Defining terms

Circle map (finding out what * What comes to mind when you students know, hear the terms 'responsible', Written think and feel, to 'global' and 'citizen'?

Difficult cases for computing the maps occur (i) when arcs of a

circle map to the ends of boundaries of elongated regions and (ii) when the circles are close to touching.

In this extended abstract, we summarize recent work relating the Poincare rotation number of a circle map [S.

It would be very interesting to relate parallel chip-firing on other graphs to iteration of a circle map (or perhaps a map on a higher-dimensional manifold) in order to explain the ubiquity of mode locking.

Hence f : R [right arrow] R descends to a circle map [bar.

Combining flamboyant Graham Bonnet-type rock vocals with purring bass riffs, robust guitar and cyclic drum patterns,

Circle map out their own terrain.

The logistic map is used to update F and CR values, while the

circle map is employed to generate chaotic generations used during search.

They are tiered lessons, cubing, graphic organizers, exit cards, learning contracts, and choice boards, and include reproducibles, directions, how the strategy fits into their

CIRCLE MAP model, examples, and comments from the teacher who developed it on its use and how students responded, with some comments from other teachers.

They consist of tiered lessons, cubing, graphic organizers, exit cards, learning contracts, and choice boards, with reproducibles, directions, examples, and discussion of how the strategy fits into their

CIRCLE MAP model, and comments from the teacher who developed it on its use and how students responded, as well as some comments from other teachers.

These are followed by six chapters on more advanced topics that include kneading theory and Hofbauer towers, detailed structure of w sets, topological entropy, lapnumbers and Markov extensions, interplay among continued fractions, and

circle maps, among others.