# David Burguet

## Title: Entropy of $C^r$, $r>1$, smooth systems via semi-algebraic tools

### Resume:

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.

# Mark Pollicott

## 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:

• Transversality;
• Differences of Cantor sets;
• $\{0,1,3\}$-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.

### Abstract:

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.

### February 13th [Monday]

Time Title Place
09.00-09.55 Registration Jabir ibn Hayyan Hall

Time Lecturer Title
10.00-11.10 Mark Pollicott Dimension and Dynamics (Background)
11.10-11.30 Coffee Break
11.30-12.40 David Burguet Entropy of $C^r$, $r>1$, smooth systems via semi-algebraic tools (I)
12.40- 14.00 Lunch
14.00-15.30 Mark Pollicott Exercise I
15.30-16.00 Coffee Break
16.00-17.30 David Burguet Exercise I

### February 14th [Tuesday]

Time Lecturer Title
10.00-11.10 Mark Pollicott Thermodynamic viewpoint and computation
11.10-11.30 Coffee Break
11.30-12.40 David Burguet Entropy of $C^r$, $r>1$, smooth systems via semi-algebraic tools (II)
12.40- 14.00 Lunch
14.00-15.30 Mark Pollicott Exercise II
15.30-16.00 Coffee Break
16.00-17.30 David Burguet Exercise II

### February 15th [Wednesday]

Time Lecturer Title
10.00-11.10 Mark Pollicott Transversality
11.10-11.30 Coffee Break
11.30-12.40 David Burguet Entropy of $C^r$, $r>1$, smooth systems via semi-algebraic tools (III)
12.40- 14.00 Lunch
14.00-15.30 Mark Pollicott Exercise III
15.30-16.00 Coffee Break
16.00-17.30 Mark Pollicott Talk at IPM

### February 16th [Thursday]

Time Lecturer Title
10.00-11.10 Mark Pollicott Microsets and Scenery flow
11.10-11.30 Coffee Break
11.30-12.40 David Burguet Entropy of $C^r$, $r>1$, smooth systems via semi-algebraic tools (IV)
12.40- 14.00 Lunch
14.00-15.30 Mark Pollicott Exercise IV
15.30-16.00 Coffee Break
16.00-17.30 David Burguet Exercise III

### February 17th [Friday]

Time Lecturer Title
10.00-11.10 Mark Pollicott Talk at seminar section Validated Numerics-Computing the dimension of $E2$ to 100 decimal places
11.10-11.30 Coffee Break
11.30-12.40 David Burguet Talk at seminar section TBA
12.40- 14.00 Lunch