Download Ergodic Theory, Hyperbolic Dynamics and Dimension Theory by Luís Barreira PDF

By Luís Barreira

Over the final 20 years, the size idea of dynamical platforms has gradually constructed into an self sustaining and intensely energetic box of study. the most objective of this quantity is to supply a unified, self-contained advent to the interaction of those 3 major parts of study: ergodic concept, hyperbolic dynamics, and size concept. It begins with the fundamental notions of the 1st subject matters and ends with a sufficiently high-level advent to the 3rd. in addition, it comprises an creation to the thermodynamic formalism, that's a huge instrument in measurement conception.

The quantity is basically meant for graduate scholars drawn to dynamical platforms, in addition to researchers in different parts who desire to find out about ergodic conception, thermodynamic formalism, or measurement idea of hyperbolic dynamics at an intermediate point in a sufficiently specific demeanour. particularly, it may be used as a foundation for graduate classes on any of those 3 matters. The textual content is also used for self-study: it really is self-contained, and apart from a few famous simple proof from different components, all statements contain targeted proofs.

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Additional resources for Ergodic Theory, Hyperbolic Dynamics and Dimension Theory

Example text

X be a measurable transformation and measure in X , then the following properties are equivalent: is a finite 1. The measure is ergodic with respect to T . 2. X; / is T -invariant almost everywhere, then it is constant almost everywhere. 2 Fourier Coefficients and Ergodicity We illustrate in this section how the Fourier coefficients can be used to establish the ergodicity of some transformations. x/ 0 for each k 2 Z. We briefly recall some properties of these numbers. '/ D ak . / for every k 2 Z.

T2 ; /. 54) for every k; ` 2 Z. '/ D 0 for every k 2 Z n f0g. '/j ¤ 0 for every k 2 Z and n 2 N. '/ Lebesgue-almost everywhere, and so ' is constant almost everywhere. This shows that the Lebesgue measure is ergodic with respect to T . For each continuous function W Œ0; 1 ! y/. 1 n kD1 kD1 Z D 'd n n lim Z T2 1 D . ˇ; / 2 R2 . This yields the desired result. 7 Applications to Number Theory 51 Fig. 2 Continued Fractions Now we consider the relation between ergodic theory and the theory of continued fractions.

0; 1/. B/ for any measurable set B t S 1 . Therefore, Eq preserves the Lebesgue measure. u Now we study the periodic points of the expanding maps of the circle. 19) 28 2 Basic Notions and Examples Fig. 4 The eight 2-periodic points of E3 Hence, the number of m-periodic points of Eq is equal to jq m 1j, and they are uniformly distributed in S 1 (see Fig. 4 for an example). In particular, each expanding map Eq has m-periodic points for every m 2 N. 5. The periodic points of an expanding map Eq are dense in S 1 , that is, the set ˚ « z 2 S 1 W z is m-periodic for Eq for some m 2 N is dense in S 1 .

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