Title: Entropy of $C^r$, $r>1$, smooth systems via semi-algebraic tools
Entropy is a master invariant in dynamical systems computing
the dynamical complexity of a system. For smooth systems Y.Yomdin has
introduced semi-algebraic tools to study continuity properties of
entropy. In this mini-course we will present some recent linked
results such as:
- uniform generators and symbolic extensions in low dimensions;
- rate of convergence of the tail entropy for $C^\infty$ smooth systems;
- equidistribution of periodic points with lower bounded Lyapunov exponents along measure of maximal entropy in low dimensions.
In the first lectures I will introduce the modern theory of entropy
structures and symbolic extensions due to Boyle and Downarowicz for
general topological systems. We will also define the asymptotic
h-expansiveness and periodic expansiveness, and then relate these
quantities to continuity of the entropy and equidistribution of
periodic points. The last lectures will be devoted to the
semi-algebraic tools and their dynamical applications to the entropy
of $C^r$, $r>1$, smooth systems.
Lecture 1: Dimension and Dynamics (Background).
This lecture will give an overview of some of the connections between fractals, dynamical systems and dimension:
- Definitions of Dimension (Box, Hausdorff, Fourier);
- Potential and measures;
- Iterated function schemes and expanding maps;
- Conformal maps and repellers;
- Examples: Julia sets and Schottky group limit sets;
- Bedford-McMullen Carpets.
Lecture 2: Thermodynamic viewpoint and computation.
This lecture will concentrate on using ideas from ergodic theory and thermodynamic formalism to give a more detailed analysis of dimension:
- Pressure and Bowen-Formula;
- Analyticity of pressure;
- Numerical approximation of dimension;
- The McMullen approach;
- The determinant approach;
- Multi fractal analysis.
Lecture 3: Transversality.
This lecture will deal with a useful approach to understanding the dimension of typical sets:
- Differences of Cantor sets;
- Bernoulli convolutions;
- Absolute continuity;
- Exceptional sets;
- (Fourier dimension).
Lecture 4: Microsets and Scenery flow.
This lecture will deal with a recent approach to studying the dimension of dynamically defined sets:
- Furstenberg's viewpoint;
- Scenary flow;
- Microsets and Measures.
Seminar: Validated Numerics-Computing the dimension of $E2$ to 100 decimal places.
The set $E2$ consists of those $0 < x < 1$ whose continued fraction expansions contain only the digits 1 or 2.
There is no known closed form expression for the dimension of $E2$. Therefore, we need to numerically estimate
the dimension using a suitably efficient algorithm, with a careful control over the error.