Dynamical Systems

Part A. 
We prove existence of smooth invariant circles for area 
preserving twist maps close enough to integrable using renormalisation. The 
smoothness depends upon that of the map and the Liouville exponent of the 
rotation number. 
 
Part B. 
Ruelle and Capocaccia gave a new definition of Gibbs states on 
Smale spaces. Equilibrium 
states of suitable function there on are known to be 
Gibbs states. The converse in discussed in this paper, where the problem is 
reduced to shift spaces and there solved by constructing suitable conjugating 
homeomorphisms in order to verify the conditions for Gibbs states which 
Bowen gave for shift spaces, where the equivalence to equilibrium states is 
known. 
 
Part C. 
On subshifts which are derived from Markov partitions exists an 
equivalence relation which idendifies points that lie on the boundary set of the 
partition. In this paper we restrict to symbolic 
dynamics. We express the 
quotient space in terms of a non-transitive subshift of 
finite type, give a 
necessary and sufficient condition for the existence of a local product 
structure and evaluate the Zeta function of the quotient space. Finally we give 
an example where the quotient space is again a subshift of finite type.

Problem 4. Fill out the following cobweb plots with starting point x 0 = 1 5 : Definition 1.2. A fixed point of a function is a point x ∈ R satisfying f (x) = x. We say that a fixed point x of a continuous function f : R → R is stable if there is some interval (a, b) containing x such that if x 0 ∈ (a, b), then the sequence x 0 , f (x 0 ), f (f (x 0 )), . . . converges to x. Otherwise it is called unstable.
Problem 7. By definition, every stable fixed point is the limit of some convergent sequence x 0 , f (x 0 ), f (f (x 0 )), . . . . Is the converse always true? That is, if x 0 , f (x 0 ), f (f (x 0 )) converges to x, must x be a stable fixed point, or at least a fixed point? Give a condition on f for when x must be a fixed point, give an example of f and x 0 where x is not a fixed point, and give an example of f and x 0 where x is a fixed point but not stable.

Logistic Maps
In this section, we study some functions called the logistic maps, which pop up in population modelling, and also have interesting mathematical properties. (Note: These are different (though related to) the logistic curve, which also pops up in population modelling problems and in calculus classes. If you've heard of that, you can think of these logistic maps as a discrete-time version of that continuous-time population model, but this discrete version has much more personality.) Let's model the size of a population. Suppose there is a species of rabbit that has a fixed generation length. If the rabbits have an infinite supply of food, then after each generation, each rabbit is replaced with r rabbits of the next generation.
1. Given this infinite supply of food, if we start with x rabbits at generation 0, how many rabbits do we have at generation t?
2. Under what circumstances will the number of rabbits approach a constant population?
Now assume that the rabbit's reproductive rate depends on the amount of available food, and that the amount of available food depends on the number of rabbits. Assume that their environment has a carrying capacity, a limit to number of rabbits that the food can support. Let's measure the population not as a natural number, counting the rabbits, but as a real number, x, which is the fraction of the carrying capacity, so that x = 0 indicates 0 rabbits, but x = 1 indicates that the population is the carrying capacity. Now assume that with each successive generation, each rabbit is replaced with r(1 − x) children, so that as the number of rabbits increases to the carrying capacity, and the amount of available food decreases to 0, the reproductive rate shrinks down from r to 0. Definition 2.1. If there are x rabbits at generation t, then there will be rx(1 − x) rabbits at generation t + 1. We call this function the logistic map with parameter r, and will use the notation 1. Describe the trends in population if 0 ≤ r < 1.
2. Explain what happens in our model if we let x at generation t be greater than 1.
3. What population at generation 0 maximizes the population at generation 1? 4. We know that our model isn't necessarily predictive if we ever have x outside the interval Problem 11. Draw a cobweb diagram for f r at r = 0.8, 1.6, 3.2, 3.5, with one or more different starting values of 0 < x < 1. In each of the diagrams, say whether or not the fixed points are stable.
Problem 12. Make a copy of the following Google Sheet: https://bit.ly/ormc-logistic. (If you're in-person, feel free to use your phone or ask an instructor to pull it up for you.) The Google Sheet calculates the same sequence x t+1 = f r (x t ) that you found in the cobweb diagram. By adjusting the value of r within the range you found in Problem 9.4, for what values of r does the population converge to a fixed point? For each such r, which fixed point does it converge to? Make sure your answer agrees with the previous examples in your cobweb plots.
The proof of this takes some work, so we put it in the bonus section.
Problem 13. For values of r between 3 and 4, when is there some regularity in what happens? When does the system become chaotic?

Optional Bonus Problems
The problems here build up the following theorem.
Theorem 2.1. For 0 ≤ r ≤ 1, 0 is the unique fixed point and is stable. For 1 ≤ r ≤ 3, 0 is unstable and 1 − 1 r is stable. For 3 < r < 4, no fixed point is stable.
Problem 14. Let f : R → R be a nonconstant polynomial. Recall that the finitely many zeroes of f split R into intervals on which f is positive and intervals on which f is negative. Show that for a fixed point x 0 to be stable, f (x) − x must be positive immediately to the left of x 0 or negative immediately to the right of x 0 .
Problem 15. Prove that 0 is a stable fixed point for 0 ≤ r ≤ 1, and that it is unstable for 1 < r ≤ 4.
We will find the following result about limits useful: Problem 16. Let a 0 , a 1 , a 2 , . . . be a sequence of numbers, let 0 ≤ r < 1, and let a be a number. Show that if for all n, |a n+1 − a| < r|a n − a|, then a 0 , a 1 , a 2 , . . . converges to a.
Problem 17. Let f (x) = ax 2 + bx + c be a quadratic, and let x 0 be a fixed point of f (x). Show that if |2ax 0 + b| < 1, then x 0 is stable. (Hint: use the previous problem.) Problem 18. Let f (x) = ax 2 + bx + c be a quadratic, and let x 0 be a fixed point of f (x). Show that if |2ax 0 + b| > 1, then x 0 is unstable.
Problem 19. Finish the proof of Theorem 2.1.

The Mandelbrot Set
The Mandelbrot set is a subset of C that you might have heard of: it's a fractal, meaning there is infinite detail as you keep zooming in.
(Image source: Wikipedia.) We will now investigate the Mandelbrot set and its connection to the logistic map. First, we recall some definitions about complex numbers.
• Recall that the complex numbers C can all be expressed as a+bi, where a, b ∈ R and i = √ −1.
• If a, b ∈ R, let a + bi = a − bi. We call this the complex conjugate of a + bi.
• If a, b ∈ R, let |a + bi| = √ a 2 + b 2 . We call this the magnitude of a + bi.
Recall that if g(x) is any bijective (one-to-one and onto) function, g −1 is the inverse of g, satisfying g −1 (g(x)) = x for all x. In other words, g −1 undoes g.
We can compute the logistic map f r (x) using the Mandelbrot map as follows: Let z = g(x) where g(x) = ax + b for some particular a and b. Then, take Mandelbrot map f c (z) for some particular c.
Finally, transform f c (z) back using g −1 , that is, f r (x) = g −1 (f c (z)). We will determine what a, b, c should be in the next question.
Similarly, we can also compute the Mandelbrot map f c (z) using the logistic map. The process is exactly the same, except that we start with g −1 . x f c Problem 22. We will investigate the connection described above.
1. Let r be given. Show that g(x) = ax + b works for some real numbers a and b. That is, solve for a, b, c in the following equation in terms of r: (1 − x)).
2. Practice using this correspondence. Let x 0 = 1 4 and r = 2. Compute x 1 , x 2 , and the corresponding z 0 , z 1 , and z 2 , using whatever method you like. Then, compute the same values a different way using the correspondence.
3. Explain why a sequence given by the logistic map x 0 , f r (x 0 ), f r (f r (x 0 )), . . . is bounded if and only if the corresponding sequence given by the Mandelbrot map z 0 , f c (z 0 ), f c (f c (z 0 )), . . . is also bounded.
We will now investigate the shape of the Mandelbrot set using this correspondence. Look back a few pages for the picture. 2. Using the correspondence from the previous question, describe the intersection of the Mandelbrot set with the real number line. Compare your answer with the picture from a few pages ago.
Next, notice that the main part of the Mandelbrot set consists of one big heart-shaped region (called a cardioid), surrounded by what looks like circles. Does each region have some kind of special property?
• For r slightly above 3, sequences converge to periodic behavior.
• For r slightly below 4, sequences exhibit chaotic behavior. Now, do the following problems. 2. Recall that from Problem 11, r = 3.2 gives periodic behavior with period 2, and r = 3.5 gives periodic behavior with period 4. What c do these correspond to? By looking at the picture of the Mandelbrot set again, can you make some more conjectures?
Problem 25. We will find a parametric equation for the boundary of the cardioid region.
1. Stable fixed points can be defined for functions f : C → C, in essentially the same way they were defined over the reals: a point satisfying f (z) = z such that for all z 0 near z, iterating z 0 , f (z 0 ), f (f (z 0 )), . . . converges to z.
Show that if r is complex, then f r (z) = rz(1−z) defined on the complex numbers has a stable fixed point if |r| < 1. (For real numbers, this means r ∈ [−1, 1]: remember that r ∈ [1, 3] also generates stable fixed points, we'll treat this case later.)