Dynamical Zeta Functions

Riemann Zeta Function Selberg Zeta function Geodesic and Anosov ﬂows Applications

Dynamical Zeta functions

Mark Pollicott

Warwick University

Chalmers’ Colloquium : 11 September, 2017

1 / 30 1 The Riemann zeta function for prime numbers (for motivation only); 2 The Selberg zeta function for lengths of closed geodesics of surfaces of constant negative curvature; and 3 The Ruelle zeta function for periods of closed orbits of geodesic and Anosov ﬂows.

In particular, the Ruelle zeta function generalises the Selberg zeta function.

The general philosophy The more we know about the zeta function the more we can deduce about the underlying properties of the numbers that are used in their deﬁnition.

The blueprint for this comes from prime numbers and the Riemann zeta function, which we brieﬂy recall.

Riemann Zeta Function Selberg Zeta function Geodesic and Anosov ﬂows Applications Diﬀerent types of zeta functions

In this talk I want to consider three types of basic zeta function.

2 / 30 2 The Selberg zeta function for lengths of closed geodesics of surfaces of constant negative curvature; and 3 The Ruelle zeta function for periods of closed orbits of geodesic and Anosov ﬂows.

In particular, the Ruelle zeta function generalises the Selberg zeta function.

The general philosophy The more we know about the zeta function the more we can deduce about the underlying properties of the numbers that are used in their deﬁnition.

The blueprint for this comes from prime numbers and the Riemann zeta function, which we brieﬂy recall.

Riemann Zeta Function Selberg Zeta function Geodesic and Anosov ﬂows Applications Diﬀerent types of zeta functions

In this talk I want to consider three types of basic zeta function.

1 The Riemann zeta function for prime numbers (for motivation only);

2 / 30 and 3 The Ruelle zeta function for periods of closed orbits of geodesic and Anosov ﬂows.

In particular, the Ruelle zeta function generalises the Selberg zeta function.

The general philosophy The more we know about the zeta function the more we can deduce about the underlying properties of the numbers that are used in their deﬁnition.

The blueprint for this comes from prime numbers and the Riemann zeta function, which we brieﬂy recall.

Riemann Zeta Function Selberg Zeta function Geodesic and Anosov ﬂows Applications Diﬀerent types of zeta functions

In this talk I want to consider three types of basic zeta function.

1 The Riemann zeta function for prime numbers (for motivation only); 2 The Selberg zeta function for lengths of closed geodesics of surfaces of constant negative curvature;

2 / 30 In particular, the Ruelle zeta function generalises the Selberg zeta function.

The general philosophy The more we know about the zeta function the more we can deduce about the underlying properties of the numbers that are used in their deﬁnition.

The blueprint for this comes from prime numbers and the Riemann zeta function, which we brieﬂy recall.

Riemann Zeta Function Selberg Zeta function Geodesic and Anosov ﬂows Applications Diﬀerent types of zeta functions

In this talk I want to consider three types of basic zeta function.

1 The Riemann zeta function for prime numbers (for motivation only); 2 The Selberg zeta function for lengths of closed geodesics of surfaces of constant negative curvature; and 3 The Ruelle zeta function for periods of closed orbits of geodesic and Anosov ﬂows.

2 / 30 The general philosophy The more we know about the zeta function the more we can deduce about the underlying properties of the numbers that are used in their deﬁnition.

The blueprint for this comes from prime numbers and the Riemann zeta function, which we brieﬂy recall.

Riemann Zeta Function Selberg Zeta function Geodesic and Anosov ﬂows Applications Diﬀerent types of zeta functions

In this talk I want to consider three types of basic zeta function.

In particular, the Ruelle zeta function generalises the Selberg zeta function.

2 / 30 The blueprint for this comes from prime numbers and the Riemann zeta function, which we brieﬂy recall.

Riemann Zeta Function Selberg Zeta function Geodesic and Anosov ﬂows Applications Diﬀerent types of zeta functions

In this talk I want to consider three types of basic zeta function.

In particular, the Ruelle zeta function generalises the Selberg zeta function.

The general philosophy The more we know about the zeta function the more we can deduce about the underlying properties of the numbers that are used in their deﬁnition.

2 / 30 Riemann Zeta Function Selberg Zeta function Geodesic and Anosov ﬂows Applications Diﬀerent types of zeta functions

In this talk I want to consider three types of basic zeta function.

In particular, the Ruelle zeta function generalises the Selberg zeta function.

The general philosophy The more we know about the zeta function the more we can deduce about the underlying properties of the numbers that are used in their deﬁnition.

The blueprint for this comes from prime numbers and the Riemann zeta function, which we brieﬂy recall.

2 / 30 (e.g., π(10) = 4, π(100) = 25, π(1000) = 168, etc.) The growth of π(x) as x → +∞ is given by the classical prime number theorem.

Theorem (Prime Number Theorem: Hadamard, de la Valle Poussin (1896))

x π(x) We have that π(x) ∼ as x → +∞, (i.e., limx→+∞ x = 1). log x log x

The main tool to prove the Prime Number Theorem is the Riemann zeta function.

Riemann Zeta Function Prime Number Theorem Selberg Zeta function The Riemann Zeta Function Geodesic and Anosov ﬂows Properties of the Riemann zeta function Applications Counting Prime Numbers

Consider the prime numbers 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, ···

Deﬁnition Given x > 0, let π(x) denote the number of primes numbers less than or equal to x.

3 / 30 π(100) = 25, π(1000) = 168, etc.) The growth of π(x) as x → +∞ is given by the classical prime number theorem.

Theorem (Prime Number Theorem: Hadamard, de la Valle Poussin (1896))

x π(x) We have that π(x) ∼ as x → +∞, (i.e., limx→+∞ x = 1). log x log x

The main tool to prove the Prime Number Theorem is the Riemann zeta function.

Riemann Zeta Function Prime Number Theorem Selberg Zeta function The Riemann Zeta Function Geodesic and Anosov ﬂows Properties of the Riemann zeta function Applications Counting Prime Numbers

Consider the prime numbers 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, ···

Deﬁnition Given x > 0, let π(x) denote the number of primes numbers less than or equal to x.

(e.g., π(10) = 4,

3 / 30 π(1000) = 168, etc.) The growth of π(x) as x → +∞ is given by the classical prime number theorem.

Theorem (Prime Number Theorem: Hadamard, de la Valle Poussin (1896))

x π(x) We have that π(x) ∼ as x → +∞, (i.e., limx→+∞ x = 1). log x log x

The main tool to prove the Prime Number Theorem is the Riemann zeta function.

Riemann Zeta Function Prime Number Theorem Selberg Zeta function The Riemann Zeta Function Geodesic and Anosov ﬂows Properties of the Riemann zeta function Applications Counting Prime Numbers

Consider the prime numbers 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, ···

Deﬁnition Given x > 0, let π(x) denote the number of primes numbers less than or equal to x.

(e.g., π(10) = 4, π(100) = 25,

3 / 30 The growth of π(x) as x → +∞ is given by the classical prime number theorem.

Theorem (Prime Number Theorem: Hadamard, de la Valle Poussin (1896))

x π(x) We have that π(x) ∼ as x → +∞, (i.e., limx→+∞ x = 1). log x log x

The main tool to prove the Prime Number Theorem is the Riemann zeta function.

Riemann Zeta Function Prime Number Theorem Selberg Zeta function The Riemann Zeta Function Geodesic and Anosov ﬂows Properties of the Riemann zeta function Applications Counting Prime Numbers

Consider the prime numbers 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, ···

Deﬁnition Given x > 0, let π(x) denote the number of primes numbers less than or equal to x.

(e.g., π(10) = 4, π(100) = 25, π(1000) = 168, etc.)

3 / 30 The main tool to prove the Prime Number Theorem is the Riemann zeta function.

Riemann Zeta Function Prime Number Theorem Selberg Zeta function The Riemann Zeta Function Geodesic and Anosov ﬂows Properties of the Riemann zeta function Applications Counting Prime Numbers

Consider the prime numbers 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, ···

Deﬁnition Given x > 0, let π(x) denote the number of primes numbers less than or equal to x.

(e.g., π(10) = 4, π(100) = 25, π(1000) = 168, etc.) The growth of π(x) as x → +∞ is given by the classical prime number theorem.

Theorem (Prime Number Theorem: Hadamard, de la Valle Poussin (1896))

x π(x) We have that π(x) ∼ as x → +∞, (i.e., limx→+∞ x = 1). log x log x

3 / 30 Riemann Zeta Function Prime Number Theorem Selberg Zeta function The Riemann Zeta Function Geodesic and Anosov ﬂows Properties of the Riemann zeta function Applications Counting Prime Numbers

Consider the prime numbers 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, ···

Deﬁnition Given x > 0, let π(x) denote the number of primes numbers less than or equal to x.

(e.g., π(10) = 4, π(100) = 25, π(1000) = 168, etc.) The growth of π(x) as x → +∞ is given by the classical prime number theorem.

Theorem (Prime Number Theorem: Hadamard, de la Valle Poussin (1896))

x π(x) We have that π(x) ∼ as x → +∞, (i.e., limx→+∞ x = 1). log x log x

The main tool to prove the Prime Number Theorem is the Riemann zeta function.

3 / 30 However, it is convenient for us to write this in the equivalent form as an Euler product Y −1 ζ(s) = 1 − p−s p where the product is over all primes p = 2, 3, 5, 7, 11, ··· .

Question What are the properties of ζ(s)?

Riemann Zeta Function Prime Number Theorem Selberg Zeta function The Riemann Zeta Function Geodesic and Anosov ﬂows Properties of the Riemann zeta function Applications The Riemann zeta function

The Riemann zeta function is the complex function (introduced by Euler) ∞ X 1 ζ(s) = ns n=1 which converges for Re(s) > 1 .

4 / 30 Question What are the properties of ζ(s)?

Riemann Zeta Function Prime Number Theorem Selberg Zeta function The Riemann Zeta Function Geodesic and Anosov ﬂows Properties of the Riemann zeta function Applications The Riemann zeta function

The Riemann zeta function is the complex function (introduced by Euler) ∞ X 1 ζ(s) = ns n=1 which converges for Re(s) > 1 . However, it is convenient for us to write this in the equivalent form as an Euler product Y −1 ζ(s) = 1 − p−s p where the product is over all primes p = 2, 3, 5, 7, 11, ··· .

4 / 30 Riemann Zeta Function Prime Number Theorem Selberg Zeta function The Riemann Zeta Function Geodesic and Anosov ﬂows Properties of the Riemann zeta function Applications The Riemann zeta function

Question What are the properties of ζ(s)?

4 / 30 Moreover, 1 ζ(s) has a simple pole at s = 1; 2 ζ(s) has no zeros on Re(s) = 1; Properties 1 and 2 are suﬃcient to prove the Prime Number Theorem. 1 ζ(s) has an analytic extension to C − {1}. The following famous conjecture was formulated by Riemann in 1859 (repeated as Hilbert’s 8th problem at the ICM-1900).

Riemann Hypothesis 1 The zeros in Re(s) > 0 lie only on the line Re(s) = 2 .

Riemann Zeta Function Prime Number Theorem Selberg Zeta function The Riemann Zeta Function Geodesic and Anosov ﬂows Properties of the Riemann zeta function Applications Properties of the Riemann zeta function

The Riemann zeta function is analytic and non-zero for Re(s) > 1.

Analytic

0 1

5 / 30 2 ζ(s) has no zeros on Re(s) = 1; Properties 1 and 2 are suﬃcient to prove the Prime Number Theorem. 1 ζ(s) has an analytic extension to C − {1}. The following famous conjecture was formulated by Riemann in 1859 (repeated as Hilbert’s 8th problem at the ICM-1900).

Riemann Hypothesis 1 The zeros in Re(s) > 0 lie only on the line Re(s) = 2 .

The Riemann zeta function is analytic and non-zero for Re(s) > 1.

Pole 0 1

Moreover, 1 ζ(s) has a simple pole at s = 1;

5 / 30 Properties 1 and 2 are suﬃcient to prove the Prime Number Theorem. 1 ζ(s) has an analytic extension to C − {1}. The following famous conjecture was formulated by Riemann in 1859 (repeated as Hilbert’s 8th problem at the ICM-1900).

Riemann Hypothesis 1 The zeros in Re(s) > 0 lie only on the line Re(s) = 2 .

The Riemann zeta function is analytic and non-zero for Re(s) > 1.

No zeros

0 1

Moreover, 1 ζ(s) has a simple pole at s = 1; 2 ζ(s) has no zeros on Re(s) = 1;

5 / 30 1 ζ(s) has an analytic extension to C − {1}. The following famous conjecture was formulated by Riemann in 1859 (repeated as Hilbert’s 8th problem at the ICM-1900).

Riemann Hypothesis 1 The zeros in Re(s) > 0 lie only on the line Re(s) = 2 .

The Riemann zeta function is analytic and non-zero for Re(s) > 1.

0 1

Moreover, 1 ζ(s) has a simple pole at s = 1; 2 ζ(s) has no zeros on Re(s) = 1; Properties 1 and 2 are suﬃcient to prove the Prime Number Theorem.

5 / 30 The following famous conjecture was formulated by Riemann in 1859 (repeated as Hilbert’s 8th problem at the ICM-1900).

Riemann Hypothesis 1 The zeros in Re(s) > 0 lie only on the line Re(s) = 2 .

The Riemann zeta function is analytic and non-zero for Re(s) > 1.

Analytic extension

0 1

Moreover, 1 ζ(s) has a simple pole at s = 1; 2 ζ(s) has no zeros on Re(s) = 1; Properties 1 and 2 are suﬃcient to prove the Prime Number Theorem. 1 ζ(s) has an analytic extension to C − {1}.

5 / 30 Riemann Zeta Function Prime Number Theorem Selberg Zeta function The Riemann Zeta Function Geodesic and Anosov ﬂows Properties of the Riemann zeta function Applications Properties of the Riemann zeta function

The Riemann zeta function is analytic and non-zero for Re(s) > 1.

0 1 1 2

Moreover, 1 ζ(s) has a simple pole at s = 1; 2 ζ(s) has no zeros on Re(s) = 1; Properties 1 and 2 are suﬃcient to prove the Prime Number Theorem. 1 ζ(s) has an analytic extension to C − {1}. The following famous conjecture was formulated by Riemann in 1859 (repeated as Hilbert’s 8th problem at the ICM-1900).

Riemann Hypothesis 1 The zeros in Re(s) > 0 lie only on the line Re(s) = 2 .

5 / 30 In summary:

Riemann Zeta Function Prime Number Theorem Selberg Zeta function The Riemann Zeta Function Geodesic and Anosov ﬂows Properties of the Riemann zeta function Applications Error terms in the Prime Number Theorem

The truth of the Riemann hypothesis would improve the Prime Number Theorem by giving an error term: Z x 1 π(x) = du + O x1/2 log x as x → +∞ 2 log u R x 1 x (where 2 log u du ∼ log x ).

6 / 30 Riemann Zeta Function Prime Number Theorem Selberg Zeta function The Riemann Zeta Function Geodesic and Anosov ﬂows Properties of the Riemann zeta function Applications Error terms in the Prime Number Theorem

The truth of the Riemann hypothesis would improve the Prime Number Theorem by giving an error term: Z x 1 π(x) = du + O x1/2 log x as x → +∞ 2 log u R x 1 x (where 2 log u du ∼ log x ). In summary:

Left board: On the top line diﬀerent forms of ζ(s) are given; and underneath the functional equation (relating ζ(s) and ζ(1 − s)) appears. Right board: The Riemann Hypothesis (asymptotic form) and the Prime Number Theorem. 6 / 30 We denote by γ one of the countably many closed geodesics on V (there is exactly one in every conjugacy class of the fundamental group and their lengths tend to inﬁnity).

Problem How many geodesics are there whose length is at most T > 0, say?

Riemann Zeta Function Deﬁnition Selberg Zeta function Properties of the Selberg zeta function Geodesic and Anosov ﬂows Error terms in the Prime Geodesic Theorem (κ = −1) Applications Prime geodesic theorems

Geometric Analogy We would like to replace prime numbers by closed geodesics on manifolds.

To begin, assume that V is a compact surface with constant curvature κ = −1.

7 / 30 Riemann Zeta Function Deﬁnition Selberg Zeta function Properties of the Selberg zeta function Geodesic and Anosov ﬂows Error terms in the Prime Geodesic Theorem (κ = −1) Applications Prime geodesic theorems

Geometric Analogy We would like to replace prime numbers by closed geodesics on manifolds.

To begin, assume that V is a compact surface with constant curvature κ = −1. We denote by γ one of the countably many closed geodesics on V (there is exactly one in every conjugacy class of the fundamental group and their lengths tend to inﬁnity).

0 Τ

Problem How many geodesics are there whose length is at most T > 0, say?

Geometric Analogy We would like to replace prime numbers by closed geodesics on manifolds.

0 Τ

Problem How many geodesics are there whose length is at most T > 0, say?

Geometric Analogy We would like to replace prime numbers by closed geodesics on manifolds.

0 Τ

Problem How many geodesics are there whose length is at most T > 0, say?

Geometric Analogy We would like to replace prime numbers by closed geodesics on manifolds.

0 Τ

Problem How many geodesics are there whose length is at most T > 0, say?

Geometric Analogy We would like to replace prime numbers by closed geodesics on manifolds.

0 Τ

Problem How many geodesics are there whose length is at most T > 0, say?

Geometric Analogy We would like to replace prime numbers by closed geodesics on manifolds.

0 Τ

Problem How many geodesics are there whose length is at most T > 0, say?

Geometric Analogy We would like to replace prime numbers by closed geodesics on manifolds.

0 Τ

Problem How many geodesics are there whose length is at most T > 0, say?

Geometric Analogy We would like to replace prime numbers by closed geodesics on manifolds.

0 Τ

Problem How many geodesics are there whose length is at most T > 0, say?

7 / 30 Deﬁnition Let Π(T ) denote the number of geodesics γ with length l(γ) less than T > 0.

The growth of Π(T ) is described by the following result

Theorem (Prime Geodesic Theorem: Selberg (1956), Huber (1959))

T We have that Π(T ) ∼ e as T → +∞, i.e., lim Π(T ) = 1. T T →+∞ eT T

Observe that this takes the same general form as the Prime Number Theorem x T π(x) ∼ log x , if one formally replaces x by e . This is traditionally proved using the appropriate zeta function... called the Selberg Zeta function.

Let γ denote a closed geodesic on V and let its length be denoted by l(γ).

8 / 30 The growth of Π(T ) is described by the following result

Theorem (Prime Geodesic Theorem: Selberg (1956), Huber (1959))

T We have that Π(T ) ∼ e as T → +∞, i.e., lim Π(T ) = 1. T T →+∞ eT T

Let γ denote a closed geodesic on V and let its length be denoted by l(γ).

Deﬁnition Let Π(T ) denote the number of geodesics γ with length l(γ) less than T > 0.

8 / 30 Observe that this takes the same general form as the Prime Number Theorem x T π(x) ∼ log x , if one formally replaces x by e . This is traditionally proved using the appropriate zeta function... called the Selberg Zeta function.

Let γ denote a closed geodesic on V and let its length be denoted by l(γ).

Deﬁnition Let Π(T ) denote the number of geodesics γ with length l(γ) less than T > 0.

The growth of Π(T ) is described by the following result

Theorem (Prime Geodesic Theorem: Selberg (1956), Huber (1959))

T We have that Π(T ) ∼ e as T → +∞, i.e., lim Π(T ) = 1. T T →+∞ eT T

8 / 30 This is traditionally proved using the appropriate zeta function... called the Selberg Zeta function.

Let γ denote a closed geodesic on V and let its length be denoted by l(γ).

Deﬁnition Let Π(T ) denote the number of geodesics γ with length l(γ) less than T > 0.

The growth of Π(T ) is described by the following result

Theorem (Prime Geodesic Theorem: Selberg (1956), Huber (1959))

T We have that Π(T ) ∼ e as T → +∞, i.e., lim Π(T ) = 1. T T →+∞ eT T

Observe that this takes the same general form as the Prime Number Theorem x T π(x) ∼ log x , if one formally replaces x by e .

8 / 30 called the Selberg Zeta function.

Let γ denote a closed geodesic on V and let its length be denoted by l(γ).

Deﬁnition Let Π(T ) denote the number of geodesics γ with length l(γ) less than T > 0.

The growth of Π(T ) is described by the following result

Theorem (Prime Geodesic Theorem: Selberg (1956), Huber (1959))

T We have that Π(T ) ∼ e as T → +∞, i.e., lim Π(T ) = 1. T T →+∞ eT T

Observe that this takes the same general form as the Prime Number Theorem x T π(x) ∼ log x , if one formally replaces x by e . This is traditionally proved using the appropriate zeta function...

8 / 30 Riemann Zeta Function Deﬁnition Selberg Zeta function Properties of the Selberg zeta function Geodesic and Anosov ﬂows Error terms in the Prime Geodesic Theorem (κ = −1) Applications Prime geodesic theorems

Let γ denote a closed geodesic on V and let its length be denoted by l(γ).

Deﬁnition Let Π(T ) denote the number of geodesics γ with length l(γ) less than T > 0.

The growth of Π(T ) is described by the following result

Theorem (Prime Geodesic Theorem: Selberg (1956), Huber (1959))

T We have that Π(T ) ∼ e as T → +∞, i.e., lim Π(T ) = 1. T T →+∞ eT T

8 / 30 Deﬁnition A version of the Selberg zeta function (actually due to Ruelle) is the complex function deﬁned by: Y −1 ζ(s) = 1 − e−sl(γ) , γ which converges to an analytic function for Re(s) > 1.

This formulation has the advantage that it looks rather similar to the Riemann zeta function where formally we replace the prime numbers p by the weights el(γ). Historically, the original Selberg zeta function was actually deﬁned by

∞ Y Y Z(s) = 1 − e−(s+n)l(γ) n=0 γ

Z(s+1) for R(s) > 1. But Z(s) and ζ(s) are easily related by ζ(s) = Z(s) .

Given a closed geodesic γ we again denote its length by l(γ).

9 / 30 This formulation has the advantage that it looks rather similar to the Riemann zeta function where formally we replace the prime numbers p by the weights el(γ). Historically, the original Selberg zeta function was actually deﬁned by

∞ Y Y Z(s) = 1 − e−(s+n)l(γ) n=0 γ

Z(s+1) for R(s) > 1. But Z(s) and ζ(s) are easily related by ζ(s) = Z(s) .

Given a closed geodesic γ we again denote its length by l(γ).

Deﬁnition A version of the Selberg zeta function (actually due to Ruelle) is the complex function deﬁned by: Y −1 ζ(s) = 1 − e−sl(γ) , γ which converges to an analytic function for Re(s) > 1.

9 / 30 Historically, the original Selberg zeta function was actually deﬁned by

∞ Y Y Z(s) = 1 − e−(s+n)l(γ) n=0 γ

Z(s+1) for R(s) > 1. But Z(s) and ζ(s) are easily related by ζ(s) = Z(s) .

Given a closed geodesic γ we again denote its length by l(γ).

This formulation has the advantage that it looks rather similar to the Riemann zeta function where formally we replace the prime numbers p by the weights el(γ).

9 / 30 Z(s+1) But Z(s) and ζ(s) are easily related by ζ(s) = Z(s) .

Given a closed geodesic γ we again denote its length by l(γ).

∞ Y Y Z(s) = 1 − e−(s+n)l(γ) n=0 γ

for R(s) > 1.

9 / 30 Riemann Zeta Function Deﬁnition Selberg Zeta function Properties of the Selberg zeta function Geodesic and Anosov ﬂows Error terms in the Prime Geodesic Theorem (κ = −1) Applications The Selberg Zeta function

Given a closed geodesic γ we again denote its length by l(γ).

∞ Y Y Z(s) = 1 − e−(s+n)l(γ) n=0 γ

Z(s+1) for R(s) > 1. But Z(s) and ζ(s) are easily related by ζ(s) = Z(s) .

9 / 30 ; and

Theorem (Selberg (1956))

ζ(s) extends meromorphically to the entire complex plane C.

Traditionally, one uses the Selberg Trace Formula to prove these resultss. This has the advantage that it gives explicitly the location of the poles and zeros in terms of the eigenvalues of the laplacian.

With this deﬁnition we have that: ζ(s) has a simple pole at s = 1; ζ(s) otherwise has no zeros or poles on Re(s) = 1

10 / 30 Traditionally, one uses the Selberg Trace Formula to prove these resultss. This has the advantage that it gives explicitly the location of the poles and zeros in terms of the eigenvalues of the laplacian.

With this deﬁnition we have that: ζ(s) has a simple pole at s = 1; ζ(s) otherwise has no zeros or poles on Re(s) = 1; and

Theorem (Selberg (1956))

ζ(s) extends meromorphically to the entire complex plane C.

10 / 30 Riemann Zeta Function Deﬁnition Selberg Zeta function Properties of the Selberg zeta function Geodesic and Anosov ﬂows Error terms in the Prime Geodesic Theorem (κ = −1) Applications Properties of the Selberg zeta function

With this deﬁnition we have that: ζ(s) has a simple pole at s = 1; ζ(s) otherwise has no zeros or poles on Re(s) = 1; and

Theorem (Selberg (1956))

ζ(s) extends meromorphically to the entire complex plane C.

10 / 30 This gives an error term in the estimate for number Π(T ) of geodesics γ with length l(γ) less than T :

Corollary

T T Z e 1 Z e 1 eT Π(T ) = du + O e(1−)T where du ∼ . 2 log u 2 log u T

We also that a weaker analogue of the “Riemann Hypothesis” does hold (due to the characterization of the zeros and poles by the spectrum of the Laplacian on V ):

Theorem (Analogue of the “Riemann Hypothesis”, Selberg (1956)) There exists > 0 such that the Selberg zeta function ζ(s) has a non-zero analytic extension to Re(s) > (1 − ) except for a simple pole at s = 1.

0 1 1 2

11 / 30 Riemann Zeta Function Deﬁnition Selberg Zeta function Properties of the Selberg zeta function Geodesic and Anosov ﬂows Error terms in the Prime Geodesic Theorem (κ = −1) Applications Error terms in the Prime Geodesic Theorem

We also that a weaker analogue of the “Riemann Hypothesis” does hold (due to the characterization of the zeros and poles by the spectrum of the Laplacian on V ):

0 1 1 2

This gives an error term in the estimate for number Π(T ) of geodesics γ with length l(γ) less than T :

Corollary

T T Z e 1 Z e 1 eT Π(T ) = du + O e(1−)T where du ∼ . 2 log u 2 log u T 11 / 30 For example, we can take a compact surface with a metric of constant curvature −1 and perturb the metric a little in some small (pink) region.

The previous approach (the Selberg trace formula) no longer applies and we need to ﬁnd a new method of proof.

Geodesic flows Riemann Zeta Function Anosov flows Selberg Zeta function Zeta Function for Anosov flows Geodesic and Anosov flows Extension for the Ruelle zeta function Applications Location of zeros Natural Problem

This brings us to a natural problem.

Problem How can we generalise these results from the speciﬁc setting of surfaces with constant curvature −1 to the much broader class of C ∞ Riemannian surfaces with (variable) negative curvature?

12 / 30 The previous approach (the Selberg trace formula) no longer applies and we need to ﬁnd a new method of proof.

This brings us to a natural problem.

For example, we can take a compact surface with a metric of constant curvature −1 and perturb the metric a little in some small (pink) region.

12 / 30 The previous approach (the Selberg trace formula) no longer applies and we need to ﬁnd a new method of proof.

This brings us to a natural problem.

For example, we can take a compact surface with a metric of constant curvature −1 and perturb the metric a little in some small (pink) region.

12 / 30 Geodesic flows Riemann Zeta Function Anosov flows Selberg Zeta function Zeta Function for Anosov flows Geodesic and Anosov flows Extension for the Ruelle zeta function Applications Location of zeros Natural Problem

This brings us to a natural problem.

For example, we can take a compact surface with a metric of constant curvature −1 and perturb the metric a little in some small (pink) region.

The previous approach (the Selberg trace formula) no longer applies and we need to ﬁnd a new method of proof.

12 / 30 Let M = {(x, v) ∈ TV : kvkx = 1} be the three dimensional space of tangent vectors of unit length (“directions”). Let φt : M → M be the geodesic ﬂow, i.e., φt (v) =γ ˙ (t) where γ : R → V is the unit speed geodesic withγ ˙ (0) = v.

Periodic orbits τ of period λ(τ) correspond to closed geodesics of length l(γ).

Aim We want to generalise the previous results to variable negative curvature manifolds.

To achieve this, we take a more dynamical point of view.

Deﬁnition (Geodesic ﬂow) Let V be a closed surface with negative curvature.

13 / 30 Let φt : M → M be the geodesic ﬂow, i.e., φt (v) =γ ˙ (t) where γ : R → V is the unit speed geodesic withγ ˙ (0) = v.

Periodic orbits τ of period λ(τ) correspond to closed geodesics of length l(γ).

Aim We want to generalise the previous results to variable negative curvature manifolds.

To achieve this, we take a more dynamical point of view.

x v

Deﬁnition (Geodesic ﬂow) Let V be a closed surface with negative curvature. Let M = {(x, v) ∈ TV : kvkx = 1} be the three dimensional space of tangent vectors of unit length (“directions”).

13 / 30 Periodic orbits τ of period λ(τ) correspond to closed geodesics of length l(γ).

Aim We want to generalise the previous results to variable negative curvature manifolds.

To achieve this, we take a more dynamical point of view.

γv

x v

Definition (Geodesic flow) Let V be a closed surface with negative curvature. Let M = {(x, v) ∈ TV : kvkx = 1} be the three dimensional space of tangent vectors of unit length (“directions”). Let φt : M → M be the geodesic flow, i.e., φt (v) =γ ˙ (t) where γ : R → V is the unit speed geodesic withγ ˙ (0) = v.

13 / 30 Periodic orbits τ of period λ(τ) correspond to closed geodesics of length l(γ).

Aim We want to generalise the previous results to variable negative curvature manifolds.

To achieve this, we take a more dynamical point of view.

φ (v) t

γv

x v

13 / 30 Periodic orbits τ of period λ(τ) correspond to closed geodesics of length l(γ).

Aim We want to generalise the previous results to variable negative curvature manifolds.

To achieve this, we take a more dynamical point of view.

x v

13 / 30 Geodesic flows Riemann Zeta Function Anosov flows Selberg Zeta function Zeta Function for Anosov flows Geodesic and Anosov flows Extension for the Ruelle zeta function Applications Location of zeros Dynamical viewpoint: Geodesic flows

Aim We want to generalise the previous results to variable negative curvature manifolds.

To achieve this, we take a more dynamical point of view.

x v= φ t (v)

Periodic orbits τ of period λ(τ) correspond to closed geodesics of length l(γ). 13 / 30 2 This work was popularized in a 1906 book by the french physicist Pierre Duhen (1861-1916); 3 In 1908 this book was translated into German by Friedrich Adler (1879-1960); 4 In 1909, Adler’s family shared a house with Albert Einstein (1879-1955) in Zurich. This translation may (or may not) have inﬂuenced Einstein’s work of general relativity; 5 On 21st October, 1916, Adler assassinated the prime minister of Austria, Count Karl von St¨urgkh(1859-1916).

14 / 30 3 In 1908 this book was translated into German by Friedrich Adler (1879-1960); 4 In 1909, Adler’s family shared a house with Albert Einstein (1879-1955) in Zurich. This translation may (or may not) have inﬂuenced Einstein’s work of general relativity; 5 On 21st October, 1916, Adler assassinated the prime minister of Austria, Count Karl von St¨urgkh(1859-1916).

14 / 30 4 In 1909, Adler’s family shared a house with Albert Einstein (1879-1955) in Zurich. This translation may (or may not) have inﬂuenced Einstein’s work of general relativity; 5 On 21st October, 1916, Adler assassinated the prime minister of Austria, Count Karl von St¨urgkh(1859-1916).

14 / 30 5 On 21st October, 1916, Adler assassinated the prime minister of Austria, Count Karl von St¨urgkh(1859-1916).

1 In 1898, geodesic ﬂows on surfaces of negative curvature and their properties were studied in a fundamental paper by Jacques Hadamard (1865-1963), only 2 years after proving the Prime Number Theorem; 2 This work was popularized in a 1906 book by the french physicist Pierre Duhen (1861-1916); 3 In 1908 this book was translated into German by Friedrich Adler (1879-1960); 4 In 1909, Adler’s family shared a house with Albert Einstein (1879-1955) in Zurich. This translation may (or may not) have inﬂuenced Einstein’s work of general relativity;

14 / 30 Geodesic flows Riemann Zeta Function Anosov flows Selberg Zeta function Zeta Function for Anosov flows Geodesic and Anosov flows Extension for the Ruelle zeta function Applications Location of zeros Historical aside: Hadamard and Geodesic Flows

14 / 30 Let τ be a closed orbit and let λ(τ) > 0 be its period ( i.e., φλ(τ)(ξ) = ξ for ξ ∈ τ). We say that the Anosov ﬂow is (topologically) weak mixing if the periods are not all integer multiples of a ﬁxed constant.

Theorem The geodesic ﬂow on a (variable) negatively curved compact surface is a weak mixing Anosov ﬂow.

More generally, an Anosov flow is a flow which transverse to the flow direction is uniformly hyperbolic, (i.e., it stretches in one direction E u and it contracts in another direction E s .)

s E

u E x φtx

Figure: (a) The hyperbolicity transverse to the orbit of an Anosov ﬂow; (b) D.V. Anosov. who once spent 3 weeks living in the guest room of my house; I did the cooking.

15 / 30 Geodesic flows Riemann Zeta Function Anosov flows Selberg Zeta function Zeta Function for Anosov flows Geodesic and Anosov flows Extension for the Ruelle zeta function Applications Location of zeros The Anosov property

More generally, an Anosov flow is a flow which transverse to the flow direction is uniformly hyperbolic, (i.e., it stretches in one direction E u and it contracts in another direction E s .)

s E

u E x φtx

Figure: (a) The hyperbolicity transverse to the orbit of an Anosov ﬂow; (b) D.V. Anosov. who once spent 3 weeks living in the guest room of my house; I did the cooking.

Let τ be a closed orbit and let λ(τ) > 0 be its period ( i.e., φλ(τ)(ξ) = ξ for ξ ∈ τ). We say that the Anosov ﬂow is (topologically) weak mixing if the periods are not all integer multiples of a ﬁxed constant.

Theorem The geodesic ﬂow on a (variable) negatively curved compact surface is a weak mixing Anosov ﬂow.

Example We can easily build a simple mechanical system which corresponds to an (Anosov) geodesic ﬂow

We can construct a model using: Two 1.5 litre bottles and two thumb tacks (for the pivots); Four drinking straws (for the rods); Six paper clips (for the joints); Thirty six penny coins (for the three masses); and Three lengths of string (to keep it all Figure: The black masses suspended in the plane) and lots of duct tape approximately on the joints consist of 12 pennies wrapped in (to hold it together). duct tape

16 / 30 Geodesic flows Riemann Zeta Function Anosov flows Selberg Zeta function Zeta Function for Anosov flows Geodesic and Anosov flows Extension for the Ruelle zeta function Applications Location of zeros The experiment

17 / 30 Let h(φ) > 0 denote the topological entropy of the ﬂow. We recall the following generalisation of the Prime Geodesic Theorem

Theorem (Prime Orbit Theorem, Margulis (1969))

eh(φ)T For weak-mixing Anosov ﬂows: Π(T ) ∼ h(φ)T as T → +∞.

The original proof of Margulis used transverse measures for the horocycle foliations and mixing arguments. However, it is possible to give a proof using zeta functions...

Denote the period of a closed orbit τ by λ(τ) > 0. Let Π(T ) denote the number of (prime) closed orbits τ with least period λ(τ) ≤ T .

18 / 30 We recall the following generalisation of the Prime Geodesic Theorem

Theorem (Prime Orbit Theorem, Margulis (1969))

eh(φ)T For weak-mixing Anosov ﬂows: Π(T ) ∼ h(φ)T as T → +∞.

The original proof of Margulis used transverse measures for the horocycle foliations and mixing arguments. However, it is possible to give a proof using zeta functions...

Denote the period of a closed orbit τ by λ(τ) > 0. Let Π(T ) denote the number of (prime) closed orbits τ with least period λ(τ) ≤ T . Let h(φ) > 0 denote the topological entropy of the ﬂow.

18 / 30 The original proof of Margulis used transverse measures for the horocycle foliations and mixing arguments. However, it is possible to give a proof using zeta functions...

Theorem (Prime Orbit Theorem, Margulis (1969))

eh(φ)T For weak-mixing Anosov ﬂows: Π(T ) ∼ h(φ)T as T → +∞.

18 / 30 Geodesic flows Riemann Zeta Function Anosov flows Selberg Zeta function Zeta Function for Anosov flows Geodesic and Anosov flows Extension for the Ruelle zeta function Applications Location of zeros The prime orbit theorem

Theorem (Prime Orbit Theorem, Margulis (1969))

eh(φ)T For weak-mixing Anosov ﬂows: Π(T ) ∼ h(φ)T as T → +∞.

The original proof of Margulis used transverse measures for the horocycle foliations and mixing arguments. However, it is possible to give a proof using zeta functions...

18 / 30 Of course, this reduces to the Selberg zeta function when the Anosov ﬂow is simply the geodesic ﬂow for a negatively curved manifold.

Recall that τ denotes a closed orbit with least period λ(τ) > 0.

Definition (Ruelle, 1975) We formally define the Ruelle zeta function for the Anosov flow to be the complex function Y −1 ζ(s) = 1 − e−sλ(τ) . τ

This converges to a non-zero analytic function for Re(s) > h(φ), where h(φ) still denotes the topological entropy of the ﬂow.

19 / 30 Geodesic flows Riemann Zeta Function Anosov flows Selberg Zeta function Zeta Function for Anosov flows Geodesic and Anosov flows Extension for the Ruelle zeta function Applications Location of zeros The zeta function for Anosov flows

Recall that τ denotes a closed orbit with least period λ(τ) > 0.

Definition (Ruelle, 1975) We formally define the Ruelle zeta function for the Anosov flow to be the complex function Y −1 ζ(s) = 1 − e−sλ(τ) . τ

This converges to a non-zero analytic function for Re(s) > h(φ), where h(φ) still denotes the topological entropy of the ﬂow.

Of course, this reduces to the Selberg zeta function when the Anosov ﬂow is simply the geodesic ﬂow for a negatively curved manifold.

19 / 30 Ruelle’s contribution was part (1) - a result that appears as an exercise in his book Thermodynamic Formalism - which Bill Parry and I couldn’t solve ....

A zeta function proof of the Prime Orbit Theorem using the Ruelle zeta function ζ(s) required only the following properties:

Theorem (Ruelle, 1978; Parry+P, 1983)

1 ζ(s) has a simple pole at s = h(φ); and 2 ζ(s) an analytic extension to a neighbourhood of {s ∈ C : Re(s) = h(φ)} − {h(φ)}.

20 / 30 - a result that appears as an exercise in his book Thermodynamic Formalism - which Bill Parry and I couldn’t solve ....

A zeta function proof of the Prime Orbit Theorem using the Ruelle zeta function ζ(s) required only the following properties:

Theorem (Ruelle, 1978; Parry+P, 1983)

1 ζ(s) has a simple pole at s = h(φ); and 2 ζ(s) an analytic extension to a neighbourhood of {s ∈ C : Re(s) = h(φ)} − {h(φ)}.

Ruelle’s contribution was part (1)

20 / 30 - which Bill Parry and I couldn’t solve ....

A zeta function proof of the Prime Orbit Theorem using the Ruelle zeta function ζ(s) required only the following properties:

Theorem (Ruelle, 1978; Parry+P, 1983)

1 ζ(s) has a simple pole at s = h(φ); and 2 ζ(s) an analytic extension to a neighbourhood of {s ∈ C : Re(s) = h(φ)} − {h(φ)}.

Ruelle’s contribution was part (1) - a result that appears as an exercise in his book Thermodynamic Formalism

20 / 30 Geodesic flows Riemann Zeta Function Anosov flows Selberg Zeta function Zeta Function for Anosov flows Geodesic and Anosov flows Extension for the Ruelle zeta function Applications Location of zeros Zeta functions and the prime orbit theorem

A zeta function proof of the Prime Orbit Theorem using the Ruelle zeta function ζ(s) required only the following properties:

Theorem (Ruelle, 1978; Parry+P, 1983)

1 ζ(s) has a simple pole at s = h(φ); and 2 ζ(s) an analytic extension to a neighbourhood of {s ∈ C : Re(s) = h(φ)} − {h(φ)}.

Ruelle’s contribution was part (1) - a result that appears as an exercise in his book Thermodynamic Formalism - which Bill Parry and I couldn’t solve ....

20 / 30 What Ruelle didn’t know is that our solution to the exercise was wrong, and so we had to use his answer after all!

In Ruelle’s article, “Dynamical Zeta Functions and Transfer Operators”, from the Notices Amer. Math. Soc. (September 2002) he recalls that he also had diﬃculty with his own exercise:

“Having obtained the above nontrivial but apparently useless result, I put it as Exercise 7(c) on page 101 in my book Thermodynamic Formalism. A few years later (December 29, 1982) Bill Parry of Warwick wrote to me about very interesting results on Axiom A ﬂows he had obtained with his student Mark Pollicott. These results used Exercise 7(c), which unfortunately he had been unable to do. Could I help? By the time I had (painfully) managed to reconstruct the solution of the exercise I received another letter: 13 Jan 83 Dear David, We’ve ﬁnally managed to do your exercise! So ignore my last letter. Sincerely, Bill Parry.”

21 / 30 Geodesic flows Riemann Zeta Function Anosov flows Selberg Zeta function Zeta Function for Anosov flows Geodesic and Anosov flows Extension for the Ruelle zeta function Applications Location of zeros Aside: Ruelle’s exercise

In Ruelle’s article, “Dynamical Zeta Functions and Transfer Operators”, from the Notices Amer. Math. Soc. (September 2002) he recalls that he also had diﬃculty with his own exercise:

21 / 30 The proof is based on writing ζ(s) using a suitable determinant of a linear operator on an appropriate Banach space of distributions. There is also an alternative proof by Dyatlov and Zworski (using a diﬀerent Banach space).

Corollary Let V be a C ∞ compact manifold with (variable) negative sectional curvatures. Then ζ(s) extends meromorphically to C.

There is an extension of the zeta function to C. Theorem (Giulietti-Liverani-P.) For a C ∞ Anosov ﬂow, the Ruelle zeta function ζ(s) has a meromorphic extension to the entire complex plane C.

22 / 30 Corollary Let V be a C ∞ compact manifold with (variable) negative sectional curvatures. Then ζ(s) extends meromorphically to C.

There is an extension of the zeta function to C. Theorem (Giulietti-Liverani-P.) For a C ∞ Anosov ﬂow, the Ruelle zeta function ζ(s) has a meromorphic extension to the entire complex plane C.

The proof is based on writing ζ(s) using a suitable determinant of a linear operator on an appropriate Banach space of distributions. There is also an alternative proof by Dyatlov and Zworski (using a diﬀerent Banach space).

22 / 30 Geodesic flows Riemann Zeta Function Anosov flows Selberg Zeta function Zeta Function for Anosov flows Geodesic and Anosov flows Extension for the Ruelle zeta function Applications Location of zeros Extension for the Ruelle zeta function

There is an extension of the zeta function to C. Theorem (Giulietti-Liverani-P.) For a C ∞ Anosov ﬂow, the Ruelle zeta function ζ(s) has a meromorphic extension to the entire complex plane C.

Corollary Let V be a C ∞ compact manifold with (variable) negative sectional curvatures. Then ζ(s) extends meromorphically to C.

22 / 30 Let V be a compact d-dimensional manifold and let h = h(φ) > 0 be the topological entropy of the geodesic ﬂow.

Theorem ( Giulietti, Liverani and P.)

1 When d ≥ 3 and V has 9 -pinched negative sectional curvatures then there exists > 0 such that ζ(s) has an analytic zero-free extension to Re(s) > h − , except for the simple pole at s = h.

As before, this leads naturally to an error term in counting closed geodesics:

Corollary 1 Assume that V has 9 -pinched negative sectional curvatures then there exists > 0 with hT Z e 1 Π(T ) = du + O(e(h−)T ). 2 log u

For d = 2 we need only assume κ < 0 (P.-Sharp (2001); after Dolgopyat (1997)).

Open Question Is the result true without any pinching condition? or for (weak mixing) Anosov ﬂows?

We also have a (weak) analogue of the Riemann Hypothesis.

23 / 30 As before, this leads naturally to an error term in counting closed geodesics:

Corollary 1 Assume that V has 9 -pinched negative sectional curvatures then there exists > 0 with hT Z e 1 Π(T ) = du + O(e(h−)T ). 2 log u

For d = 2 we need only assume κ < 0 (P.-Sharp (2001); after Dolgopyat (1997)).

Open Question Is the result true without any pinching condition? or for (weak mixing) Anosov ﬂows?

We also have a (weak) analogue of the Riemann Hypothesis. Let V be a compact d-dimensional manifold and let h = h(φ) > 0 be the topological entropy of the geodesic ﬂow.

Theorem ( Giulietti, Liverani and P.)

1 When d ≥ 3 and V has 9 -pinched negative sectional curvatures then there exists > 0 such that ζ(s) has an analytic zero-free extension to Re(s) > h − , except for the simple pole at s = h.

23 / 30 For d = 2 we need only assume κ < 0 (P.-Sharp (2001); after Dolgopyat (1997)).

Open Question Is the result true without any pinching condition? or for (weak mixing) Anosov ﬂows?

We also have a (weak) analogue of the Riemann Hypothesis. Let V be a compact d-dimensional manifold and let h = h(φ) > 0 be the topological entropy of the geodesic ﬂow.

Theorem ( Giulietti, Liverani and P.)

1 When d ≥ 3 and V has 9 -pinched negative sectional curvatures then there exists > 0 such that ζ(s) has an analytic zero-free extension to Re(s) > h − , except for the simple pole at s = h.

As before, this leads naturally to an error term in counting closed geodesics:

Corollary 1 Assume that V has 9 -pinched negative sectional curvatures then there exists > 0 with hT Z e 1 Π(T ) = du + O(e(h−)T ). 2 log u

23 / 30 Open Question Is the result true without any pinching condition? or for (weak mixing) Anosov ﬂows?

We also have a (weak) analogue of the Riemann Hypothesis. Let V be a compact d-dimensional manifold and let h = h(φ) > 0 be the topological entropy of the geodesic ﬂow.

Theorem ( Giulietti, Liverani and P.)

1 When d ≥ 3 and V has 9 -pinched negative sectional curvatures then there exists > 0 such that ζ(s) has an analytic zero-free extension to Re(s) > h − , except for the simple pole at s = h.

As before, this leads naturally to an error term in counting closed geodesics:

Corollary 1 Assume that V has 9 -pinched negative sectional curvatures then there exists > 0 with hT Z e 1 Π(T ) = du + O(e(h−)T ). 2 log u

For d = 2 we need only assume κ < 0 (P.-Sharp (2001); after Dolgopyat (1997)).

23 / 30 or for (weak mixing) Anosov ﬂows?

We also have a (weak) analogue of the Riemann Hypothesis. Let V be a compact d-dimensional manifold and let h = h(φ) > 0 be the topological entropy of the geodesic ﬂow.

Theorem ( Giulietti, Liverani and P.)

1 When d ≥ 3 and V has 9 -pinched negative sectional curvatures then there exists > 0 such that ζ(s) has an analytic zero-free extension to Re(s) > h − , except for the simple pole at s = h.

As before, this leads naturally to an error term in counting closed geodesics:

Corollary 1 Assume that V has 9 -pinched negative sectional curvatures then there exists > 0 with hT Z e 1 Π(T ) = du + O(e(h−)T ). 2 log u

For d = 2 we need only assume κ < 0 (P.-Sharp (2001); after Dolgopyat (1997)).

Open Question Is the result true without any pinching condition?

23 / 30 Geodesic flows Riemann Zeta Function Anosov flows Selberg Zeta function Zeta Function for Anosov flows Geodesic and Anosov flows Extension for the Ruelle zeta function Applications Location of zeros Location of zeros

We also have a (weak) analogue of the Riemann Hypothesis. Let V be a compact d-dimensional manifold and let h = h(φ) > 0 be the topological entropy of the geodesic ﬂow.

Theorem ( Giulietti, Liverani and P.)

1 When d ≥ 3 and V has 9 -pinched negative sectional curvatures then there exists > 0 such that ζ(s) has an analytic zero-free extension to Re(s) > h − , except for the simple pole at s = h.

As before, this leads naturally to an error term in counting closed geodesics:

Corollary 1 Assume that V has 9 -pinched negative sectional curvatures then there exists > 0 with hT Z e 1 Π(T ) = du + O(e(h−)T ). 2 log u

For d = 2 we need only assume κ < 0 (P.-Sharp (2001); after Dolgopyat (1997)).

Open Question Is the result true without any pinching condition? or for (weak mixing) Anosov ﬂows?

23 / 30 In this case the pole occurs at s = dimH (X ) Example Let T : J → J be a hyperbolic Julia set.

Figure: (i) A Julia set; (ii) M.P. at the grave of Felix Hausdorﬀ

Riemann Zeta Function Selberg Zeta function 1. Computing Hausdorff Dimension Geodesic and Anosov flows 2. Exponential Decay of correlations Applications Application 1: Computing Hausdorff Dimension

We can associate to a real analytic conformal expanding map T : X → X an analogous zeta function where the lengths of closed orbits λ(τ) are replaced by weights log |(T n)0(x)| for periodic orbits T nx = x, i.e., ∞ ! X 1 X ζ(s) = exp |(T n)0(x)|−s n n=1 T nx=x

24 / 30 Riemann Zeta Function Selberg Zeta function 1. Computing Hausdorff Dimension Geodesic and Anosov flows 2. Exponential Decay of correlations Applications Application 1: Computing Hausdorff Dimension

In this case the pole occurs at s = dimH (X ) Example Let T : J → J be a hyperbolic Julia set.

Figure: (i) A Julia set; (ii) M.P. at the grave of Felix Hausdorﬀ

24 / 30 Example Let X ⊂ [0, 1] be the nonlinear Cantor set whose continued fraction expansion contains only the digits 1 or 2. 1 1 The usual Gauss map T : X → X given by T (x) = x − x is expanding and real analytic.

There is no closed form expression for dimH (X ) but when N = 25, say, then Jenkinson-P. numerically estimated (with an accuracy of 10−102):

dimH (E2) = 0.531280506277205141624468647368471785 493059109018398779888397803927529535 6438313459181095701811852398 ···

Previous best was 15 decimal places way back in 2016.

Riemann Zeta Function Selberg Zeta function 1. Computing Hausdorff Dimension Geodesic and Anosov flows 2. Exponential Decay of correlations Applications Computing Hausdorff Dimension

Following ideas of Ruelle and Grothendieck (describing ζ(s) in terms of the determinant of a nuclear operator) the zeta function ζ(s), and thus its pole dimH (X ) can be estimated very rapidly in terms of the weights {|(T n)0(x)| : T nx = x for n ≤ N} for any N > 0.

25 / 30 1 1 The usual Gauss map T : X → X given by T (x) = x − x is expanding and real analytic.

There is no closed form expression for dimH (X ) but when N = 25, say, then Jenkinson-P. numerically estimated (with an accuracy of 10−102):

dimH (E2) = 0.531280506277205141624468647368471785 493059109018398779888397803927529535 6438313459181095701811852398 ···

Previous best was 15 decimal places way back in 2016.

Riemann Zeta Function Selberg Zeta function 1. Computing Hausdorff Dimension Geodesic and Anosov flows 2. Exponential Decay of correlations Applications Computing Hausdorff Dimension

Example Let X ⊂ [0, 1] be the nonlinear Cantor set whose continued fraction expansion contains only the digits 1 or 2.

25 / 30 There is no closed form expression for dimH (X ) but when N = 25, say, then Jenkinson-P. numerically estimated (with an accuracy of 10−102):

dimH (E2) = 0.531280506277205141624468647368471785 493059109018398779888397803927529535 6438313459181095701811852398 ···

Previous best was 15 decimal places way back in 2016.

Riemann Zeta Function Selberg Zeta function 1. Computing Hausdorff Dimension Geodesic and Anosov flows 2. Exponential Decay of correlations Applications Computing Hausdorff Dimension

Example Let X ⊂ [0, 1] be the nonlinear Cantor set whose continued fraction expansion contains only the digits 1 or 2. 1 1 The usual Gauss map T : X → X given by T (x) = x − x is expanding and real analytic.

25 / 30 Previous best was 15 decimal places way back in 2016.

Riemann Zeta Function Selberg Zeta function 1. Computing Hausdorff Dimension Geodesic and Anosov flows 2. Exponential Decay of correlations Applications Computing Hausdorff Dimension

There is no closed form expression for dimH (X ) but when N = 25, say, then Jenkinson-P. numerically estimated (with an accuracy of 10−102):

dimH (E2) = 0.531280506277205141624468647368471785 493059109018398779888397803927529535 6438313459181095701811852398 ···

25 / 30 Riemann Zeta Function Selberg Zeta function 1. Computing Hausdorff Dimension Geodesic and Anosov flows 2. Exponential Decay of correlations Applications Computing Hausdorff Dimension

There is no closed form expression for dimH (X ) but when N = 25, say, then Jenkinson-P. numerically estimated (with an accuracy of 10−102):

dimH (E2) = 0.531280506277205141624468647368471785 493059109018398779888397803927529535 6438313459181095701811852398 ···

Previous best was 15 decimal places way back in 2016.

25 / 30 We say the ﬂow is mixing if ρ(t) → 0 as t → +∞, for all F , G ∈ C ∞(M). We say the ﬂow is exponentially mixing if ∃λ > 0 ∀F , G ∈ C ∞(M) ∃C > 0 such that |ρ(t)| ≤ Ce−λ|t|.

If we consider geodesic ﬂows, then as before we have classical results when the underlying manifold V has constant curvature.

Theorem (“Decay of matrix coeﬃcients”)

If V has constant negative (sectional) curvatures and µLiou is the normalised Liouville/Haar measure then we have exponential mixing.

Riemann Zeta Function Selberg Zeta function 1. Computing Hausdorff Dimension Geodesic and Anosov flows 2. Exponential Decay of correlations Applications 2. Exponential Decay of correlations for geodesic flows

Deﬁnition ∞ ∞ We can associate to the C ﬂow φt : M → M, a probability measure µ and the C functions F , G : M → R a correlation function: Z Z Z ρ(t) := F ◦ φt · Gdµ − Fdµ Gdµ, for t ∈ R.

26 / 30 We say the ﬂow is exponentially mixing if ∃λ > 0 ∀F , G ∈ C ∞(M) ∃C > 0 such that |ρ(t)| ≤ Ce−λ|t|.

If we consider geodesic ﬂows, then as before we have classical results when the underlying manifold V has constant curvature.

Theorem (“Decay of matrix coeﬃcients”)

If V has constant negative (sectional) curvatures and µLiou is the normalised Liouville/Haar measure then we have exponential mixing.

We say the ﬂow is mixing if ρ(t) → 0 as t → +∞, for all F , G ∈ C ∞(M).

26 / 30 If we consider geodesic ﬂows, then as before we have classical results when the underlying manifold V has constant curvature.

Theorem (“Decay of matrix coeﬃcients”)

If V has constant negative (sectional) curvatures and µLiou is the normalised Liouville/Haar measure then we have exponential mixing.

We say the ﬂow is mixing if ρ(t) → 0 as t → +∞, for all F , G ∈ C ∞(M). We say the ﬂow is exponentially mixing if ∃λ > 0 ∀F , G ∈ C ∞(M) ∃C > 0 such that |ρ(t)| ≤ Ce−λ|t|.

26 / 30 Riemann Zeta Function Selberg Zeta function 1. Computing Hausdorff Dimension Geodesic and Anosov flows 2. Exponential Decay of correlations Applications 2. Exponential Decay of correlations for geodesic flows

If we consider geodesic ﬂows, then as before we have classical results when the underlying manifold V has constant curvature.

Theorem (“Decay of matrix coeﬃcients”)

If V has constant negative (sectional) curvatures and µLiou is the normalised Liouville/Haar measure then we have exponential mixing.

26 / 30 Let us begin with the speciﬁc choice that µ = µLiou is the Liouville measure (volume) on M.

Theorem (Dolgopyat, Liverani)

For the Liouville measure µLiou the geodesic ﬂow mixes exponentially fast

Dolgopyat (1998) originally proved the result assuming that the sectional 1 curvatures are 4 -pinched; Liverani (2004) extended the method to any negative sectional curvatures.

Riemann Zeta Function Selberg Zeta function 1. Computing Hausdorff Dimension Geodesic and Anosov flows 2. Exponential Decay of correlations Applications Variable curvature geodesic flows: Liouville measure

Question Can one prove exponential mixing in the more general setting of variable negative curvature geodesic ﬂows?

27 / 30 Dolgopyat (1998) originally proved the result assuming that the sectional 1 curvatures are 4 -pinched; Liverani (2004) extended the method to any negative sectional curvatures.

Question Can one prove exponential mixing in the more general setting of variable negative curvature geodesic ﬂows?

Let us begin with the speciﬁc choice that µ = µLiou is the Liouville measure (volume) on M.

Theorem (Dolgopyat, Liverani)

For the Liouville measure µLiou the geodesic ﬂow mixes exponentially fast

27 / 30 Riemann Zeta Function Selberg Zeta function 1. Computing Hausdorff Dimension Geodesic and Anosov flows 2. Exponential Decay of correlations Applications Variable curvature geodesic flows: Liouville measure

Question Can one prove exponential mixing in the more general setting of variable negative curvature geodesic ﬂows?

Let us begin with the speciﬁc choice that µ = µLiou is the Liouville measure (volume) on M.

Theorem (Dolgopyat, Liverani)

For the Liouville measure µLiou the geodesic ﬂow mixes exponentially fast

Dolgopyat (1998) originally proved the result assuming that the sectional 1 curvatures are 4 -pinched; Liverani (2004) extended the method to any negative sectional curvatures.

27 / 30 The connection with the zeta functions is the following.

Lemma R ∞ ist The poles for the Fourier transform ρb(s) = −∞ e ρ(t)dt of the correlation function are determined by those for ζ(s) (i.e., translate h(φ) to 0, turn the plane through 90 degrees, and add its reﬂection in the real line.)

The previous results for ζ(s) and the Paley-Wiener theorem show the following:

Corollary (Giulietti-Liverani-P., 2013)

1 Assume that the negative sectional curvatures of V are 9 -pinched. Then for the Bowen-Margulis measure µBM the geodesic ﬂow mixes exponentially fast.

Again, for d = 2 we need only assume κ < 0 (Dolgopyat).

Question Is the result true (for d ≥ 3) without the additional condition on the curvature?

Riemann Zeta Function Selberg Zeta function 1. Computing Hausdorﬀ Dimension Geodesic and Anosov ﬂows 2. Exponential Decay of correlations Applications Exponential mixing for the Bowen-Margulis measure

We can try to extend these results to the Bowen-Margulis measure (or measure of maximal entropy), say.

28 / 30 The previous results for ζ(s) and the Paley-Wiener theorem show the following:

Corollary (Giulietti-Liverani-P., 2013)

1 Assume that the negative sectional curvatures of V are 9 -pinched. Then for the Bowen-Margulis measure µBM the geodesic ﬂow mixes exponentially fast.

Again, for d = 2 we need only assume κ < 0 (Dolgopyat).

Question Is the result true (for d ≥ 3) without the additional condition on the curvature?

We can try to extend these results to the Bowen-Margulis measure (or measure of maximal entropy), say. The connection with the zeta functions is the following.

ε 0 0 1 ε

28 / 30 Again, for d = 2 we need only assume κ < 0 (Dolgopyat).

Question Is the result true (for d ≥ 3) without the additional condition on the curvature?

We can try to extend these results to the Bowen-Margulis measure (or measure of maximal entropy), say. The connection with the zeta functions is the following.

ε 0 0 1 ε

ε The previous results for ζ(s) and the Paley-Wiener theorem show the following:

Corollary (Giulietti-Liverani-P., 2013)

1 Assume that the negative sectional curvatures of V are 9 -pinched. Then for the Bowen-Margulis measure µBM the geodesic ﬂow mixes exponentially fast.

28 / 30 Riemann Zeta Function Selberg Zeta function 1. Computing Hausdorﬀ Dimension Geodesic and Anosov ﬂows 2. Exponential Decay of correlations Applications Exponential mixing for the Bowen-Margulis measure

We can try to extend these results to the Bowen-Margulis measure (or measure of maximal entropy), say. The connection with the zeta functions is the following.

The previous results for ζ(s) and the Paley-Wiener theorem show the following:

Corollary (Giulietti-Liverani-P., 2013)

1 Assume that the negative sectional curvatures of V are 9 -pinched. Then for the Bowen-Margulis measure µBM the geodesic ﬂow mixes exponentially fast.

Again, for d = 2 we need only assume κ < 0 (Dolgopyat).

Question Is the result true (for d ≥ 3) without the additional condition on the curvature?

28 / 30 Finally, we can ask whether there are similar results for other Gibbs measures µGibbs associated to H¨olderpotentials ψ : M → R? Theorem Relative to a Gibbs measure the ﬂow mixes exponentially fast, providing the sectional 1 curvatures satisfy −1 < κ < − 4 .

This result was claimed in the preprint of Dolgopyat’s original paper, but the final version required additionally the doubling property (i.e., ∃C > 0, ∀x ∈ M, ∀r > 0, µGibbs (B(x, 2r)) ≤ CµGibbs (B(x, r))). However, this can be shown for geodesic flows (using the Mohsen Shadowing Lemma, cf. forthcoming Astérisquebook of Paulin, P., and Schapira).

Usual Question Is the result still true without the pinching condition on the curvature?

Riemann Zeta Function Selberg Zeta function 1. Computing Hausdorﬀ Dimension Geodesic and Anosov ﬂows 2. Exponential Decay of correlations Applications Exponential mixing for other Gibbs measures

Application The mixing result for the Bowen-Margulis measure can in turn be use to get error terms for other counting problems (e.g., counting geodesic arcs in homotopy classes).

29 / 30 This result was claimed in the preprint of Dolgopyat’s original paper, but the final version required additionally the doubling property (i.e., ∃C > 0, ∀x ∈ M, ∀r > 0, µGibbs (B(x, 2r)) ≤ CµGibbs (B(x, r))). However, this can be shown for geodesic flows (using the Mohsen Shadowing Lemma, cf. forthcoming Astérisquebook of Paulin, P., and Schapira).

Usual Question Is the result still true without the pinching condition on the curvature?

Riemann Zeta Function Selberg Zeta function 1. Computing Hausdorﬀ Dimension Geodesic and Anosov ﬂows 2. Exponential Decay of correlations Applications Exponential mixing for other Gibbs measures

Application The mixing result for the Bowen-Margulis measure can in turn be use to get error terms for other counting problems (e.g., counting geodesic arcs in homotopy classes).

Finally, we can ask whether there are similar results for other Gibbs measures µGibbs associated to H¨olderpotentials ψ : M → R? Theorem Relative to a Gibbs measure the ﬂow mixes exponentially fast, providing the sectional 1 curvatures satisfy −1 < κ < − 4 .

29 / 30 However, this can be shown for geodesic ﬂows (using the Mohsen Shadowing Lemma, cf. forthcoming Ast´erisquebook of Paulin, P., and Schapira).

Usual Question Is the result still true without the pinching condition on the curvature?

Riemann Zeta Function Selberg Zeta function 1. Computing Hausdorﬀ Dimension Geodesic and Anosov ﬂows 2. Exponential Decay of correlations Applications Exponential mixing for other Gibbs measures

Application The mixing result for the Bowen-Margulis measure can in turn be use to get error terms for other counting problems (e.g., counting geodesic arcs in homotopy classes).

29 / 30 Usual Question Is the result still true without the pinching condition on the curvature?

Riemann Zeta Function Selberg Zeta function 1. Computing Hausdorﬀ Dimension Geodesic and Anosov ﬂows 2. Exponential Decay of correlations Applications Exponential mixing for other Gibbs measures

Application The mixing result for the Bowen-Margulis measure can in turn be use to get error terms for other counting problems (e.g., counting geodesic arcs in homotopy classes).

29 / 30 Riemann Zeta Function Selberg Zeta function 1. Computing Hausdorﬀ Dimension Geodesic and Anosov ﬂows 2. Exponential Decay of correlations Applications Exponential mixing for other Gibbs measures

Application The mixing result for the Bowen-Margulis measure can in turn be use to get error terms for other counting problems (e.g., counting geodesic arcs in homotopy classes).

Usual Question Is the result still true without the pinching condition on the curvature?

29 / 30 Riemann Zeta Function Selberg Zeta function 1. Computing Hausdorﬀ Dimension Geodesic and Anosov ﬂows 2. Exponential Decay of correlations Applications The End

Thank you for your attention

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