By Yakov Pesin and Vaughn Climenhaga

ISBN-10: 0821848895

ISBN-13: 9780821848890

Either fractal geometry and dynamical structures have a protracted background of improvement and feature supplied fertile flooring for lots of nice mathematicians and masses deep and critical arithmetic. those components have interaction with one another and with the speculation of chaos in a primary method: many dynamical platforms (even a few extremely simple ones) produce fractal units, that are in flip a resource of abnormal ``chaotic'' motions within the procedure. This ebook is an creation to those fields, with an emphasis at the courting among them. the 1st 1/2 the e-book introduces a number of the key principles in fractal geometry and size theory--Cantor units, Hausdorff size, field dimension--using dynamical notions every time attainable, relatively one-dimensional Markov maps and symbolic dynamics. a variety of options for computing Hausdorff measurement are proven, resulting in a dialogue of Bernoulli and Markov measures and of the connection among size, entropy, and Lyapunov exponents. within the moment half the booklet a few examples of dynamical structures are thought of and numerous phenomena of chaotic behaviour are mentioned, together with bifurcations, hyperbolicity, attractors, horseshoes, and intermittent and chronic chaos. those phenomena are certainly printed during our examine of 2 actual versions from science--the FitzHugh-Nagumo version and the Lorenz procedure of differential equations. This booklet is obtainable to undergraduate scholars and calls for merely general wisdom in calculus, linear algebra, and differential equations. components of aspect set topology and degree thought are brought as wanted. This booklet is because of the the MASS direction in research at Penn nation college within the fall semester of 2008

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I Each partial sum over ﬁnitely many disjoint intervals is less than 1, and so the inﬁnite sum converges. 16) to sets which are not countable unions of intervals, we cover them with such unions. 17) Leb(Ii ) {Ii } is a cover of Z by open intervals . Leb(Z) = inf i In fact, when we give a less superﬁcial treatment of measure theory in Chapter 3, we will see that what we have just deﬁned is actually an outer measure, unless we require Z to be a measurable set. For the time being, though, this rough idea will serve us well as motivation for other deﬁnitions and discussions, and we will return to the technical details in due course.

To address this shortcoming, we introduce a new term into our equation. Suppose environmental factors determine some maximum population P , which corresponds, for instance, to the amount of food (or some other resource) available. Then the population cannot grow beyond P ; furthermore, if the population reaches P , all the food will be eaten and the platypi will starve, sending the next year’s Lecture 2 17 population to 0. We model this situation with the formula x → rx(P − x). In order to keep the equations as simple as possible, though, we consider the quantity x ˜ = x/P , which stands for the proportion of the maximum population P , and so lies between 0 and 1.

The correspondence runs deeper than that, to an equivalence between the dynamics of the two sets as well. Of course, at this point we have not put any dynamics on the + + set Σ+ 2 , and so we must deﬁne a map σ : Σ2 → Σ2 in order for the previous claim to make any sense. 9) f (h(w)) = Iw2 ∩ f −1 (Iw3 ) ∩ f −2 (Iw4 ) ∩ · · · = h(w ) where we write w = (w2 , w3 , . . 12 The map which takes w to w is particularly simple: all we have to 11We will see later that cardinality is a zero-dimensional measure, and so this result says that C is somehow “bigger than zero-dimensional”.

### Lectures on fractal geometry and dynamical systems by Yakov Pesin and Vaughn Climenhaga

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