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# Documents  Pollicott, Mark | enregistrements trouvés : 18

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## Interview at CIRM: Mark Pollicott Pollicott, Mark | CIRM H

Post-edited

Outreach;Mathematics Education and Popularization of Mathematics

Mark Pollicott (born 24 September 1959) is a British mathematician known for his contributions to ergodic theory and dynamical systems. He has a particular interest in applications to other areas of mathematics, including geometry, number theory and analysis.

Pollicott attended High Pavement College in Nottingham, where his teachers included the Booker prize winning author Stanley Middleton. He gained a BSc in Mathematics and Physics in 1981 and a PhD in Mathematics in 1984 both at the University of Warwick. His PhD supervisor was Bill Parry and his thesis title The Ruelle Operator, Zeta Functions and the Asymptotic Distribution of Closed Orbits.

He held permanent positions at the University of Edinburgh, University of Porto, and University of Warwick before appointment to the Fielden Chair of Pure Mathematics in Manchester (1996-2004). He then returned to a professorship at Warwick in 2005. In addition, he has held numerous visiting positions including ones at the IHES in Paris, the Institute for Advanced Study in Princeton, MSRI in University of California, Berkeley, Caltech and Grenoble. He has been recipient of a Royal Society University Research Fellowship, two Leverhulme Trust Senior Research Fellowships and an E.U. Marie Curie Chair.
Mark Pollicott (born 24 September 1959) is a British mathematician known for his contributions to ergodic theory and dynamical systems. He has a particular interest in applications to other areas of mathematics, including geometry, number theory and analysis.

Pollicott attended High Pavement College in Nottingham, where his teachers included the Booker prize winning author Stanley Middleton. He gained a BSc in Mathematics and Physics in 1981 ...

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## Emergence of wandering stable components Berger, Pierre | CIRM H

Post-edited

Research talks;Dynamical Systems and Ordinary Differential Equations

In a joint work with Sebastien Biebler, we show the existence of a locally dense set of real polynomial automorphisms of $\mathbb{C}^{2}$ displaying a stable wandering Fatou component; in particular this solves the problem of their existence, reported by Bedford and Smillie in 1991. These wandering Fatou components have non-empty real trace and their statistical behavior is historical with high emergence. The proof follows from a real geometrical model which enables us to show the existence of an open and dense set of $C^{r}$ families of surface diffeomorphisms in the Newhouse domain, each of which displaying a historical, high emergent, wandering domain at a dense set of parameters, for every $2\leq r\leq \infty$ and $r=\omega$. Hence, this also complements the recent work of Kiriki and Soma, by proving the last Taken's problem in the $C^{\infty }$ and $C^{\omega }$-case. In a joint work with Sebastien Biebler, we show the existence of a locally dense set of real polynomial automorphisms of $\mathbb{C}^{2}$ displaying a stable wandering Fatou component; in particular this solves the problem of their existence, reported by Bedford and Smillie in 1991. These wandering Fatou components have non-empty real trace and their statistical behavior is historical with high emergence. The proof follows from a real g...

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## Ergodic theory of Zd actions :proceedings of the Warwick symposium#1993-1994 Pollicott, Mark ; Schmidt, Klaus D. | Cambridge University Press 1996

Congrès

- 484 p.
ISBN 978-0-521-57688-8

London mathematical society lecture note series , 0228

Localisation : Collection 1er étage

Zelta action # Zeta d action # Zeta réaction # automorphisme de groupe compact # dimension de Hausdorff # flux sur espace homogène # invariant K-théorique # principe variationnel # propriété de rigidité # système dynamique # théorie de Ramsey # théorie des nombres combinatoire # théorie ergodique

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## Unique equilibrium states for geodesic flows over manifolds without focal-points Kao, Lien-Yung | CIRM H

Multi angle

Research School;Dynamical Systems and Ordinary Differential Equations;Geometry

We study dynamics of geodesic flows over closed surfaces of genus greater than or equal to 2 without focal points. Especially, we prove that there is a large class of potentials having unique equilibrium states, including scalar multiples of the geometric potential, provided the scalar is less than 1. Moreover, we discuss ergodic properties of these unique equilibrium states. We show these unique equilibrium states are Bernoulli, and weighted regular periodic orbits are equidistributed relative to these unique equilibrium states. We study dynamics of geodesic flows over closed surfaces of genus greater than or equal to 2 without focal points. Especially, we prove that there is a large class of potentials having unique equilibrium states, including scalar multiples of the geometric potential, provided the scalar is less than 1. Moreover, we discuss ergodic properties of these unique equilibrium states. We show these unique equilibrium states are Bernoulli, and weighted ...

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## Thermodynamic formalism in transcendental meromorphic dynamics Urbanski, Mariusz | CIRM H

Multi angle

Research School;Dynamical Systems and Ordinary Differential Equations

I will present some joint works with Volker Mayer in which we primarily show that for a large class of entire and meromorphic transcendental functions the full geometric thermodynamic formalism holds. Most notably, this means that the transfer operators generated by geometric potentials are well dened and bounded after an appropriate conformal change of Riemannian metric on the complex plane C. We show that these operators are quasi-compact of diagonal type with one leading eigenvalue, which in addition is simple. In particular, the dual operators have positive eigenvalues and eigenvectors that are Borel probability eigenmeasures. The probability measure obtained by integrating these eigenmeasures against leading eigenfanctions of transfer operators are invariant. We show that these measures are equilibrium states of geometric potentials. The primary applications of these theorems capture the stochastic laws such as exponential decay of correlations, the central limit theorem, and the law of iterated logarithm. it also permits us to provide exact formulas (of Bowen’s type) for Hausdorff dimension of radial Julia sets and multifractal analysis. We will discuss two distinct routes (leading to different though overlapping classes of meromorphic transcendental functions) to get the geometric thermodynamic formalism. One of them is based on Nevanlina’s theory and the other on analogues of integral means spectrum from classical complex analysis of conformal maps. I will present some joint works with Volker Mayer in which we primarily show that for a large class of entire and meromorphic transcendental functions the full geometric thermodynamic formalism holds. Most notably, this means that the transfer operators generated by geometric potentials are well dened and bounded after an appropriate conformal change of Riemannian metric on the complex plane C. We show that these operators are quasi-compact of ...

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## Multistability of the climate : noise-induced transitions across Melancholia states, invariant measure, and phase transition Lucarini, Valerio | CIRM H

Multi angle

Research School;Dynamical Systems and Ordinary Differential Equations;Mathematical Physics

For a wide range of values of the incoming solar radiation, the Earth features at least two attracting states, which correspond to competing climates. The warm climate is analogous to the present one, the snowball climate features global glaciation and conditions that can hardly support life forms. Paleoclimatic evidences suggest that in past our planet flipped between these two states. The main physical mechanism responsible for such instability is the ice-albedo feedback. In a previous work, we defined the Melancholia states that sit between the two climates. Such states are embedded in the boundaries between the two basins of attraction and feature extensive glaciation down to relatively low latitudes. Here, we explore the global stability properties of the system by introducing random perturbations as modulations to the intensity of the incoming solar radiation. We observe noise-induced transitions between the competing basins of attractions. In the weak noise limit, large deviation laws define the invariant measure and the statistics of escape times. By empirically constructing the instantons, we show that the Melancholia states are the gateways for the noise-induced transitions. In the region of multistability, in the zero-noise limit, the measure is supported only on one of the competing attractors. For low (high) values of the solar irradiance, the limit measure is the snowball (warm) climate. The changeover between the two regimes corresponds to a first order phase transition in the system. The framework we propose seems of general relevance for the study of complex multistable systems. At this regard, we relate our results to the debate around the prominence of contigency vs. convergence in biological evolution. Finally, we propose a new method for constructing Melancholia states from direct numerical simulations, thus bypassing the need to use the edge-tracking algorithm. For a wide range of values of the incoming solar radiation, the Earth features at least two attracting states, which correspond to competing climates. The warm climate is analogous to the present one, the snowball climate features global glaciation and conditions that can hardly support life forms. Paleoclimatic evidences suggest that in past our planet flipped between these two states. The main physical mechanism responsible for such i...

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## Marked length spectrum rigidity and the geodesic stretch Knieper, Gerhard | CIRM H

Multi angle

Research talks;Dynamical Systems and Ordinary Differential Equations

Joint work with Guillarmou and Lefeuvre.

37D40

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## Linear and fractional response: a survey Baladi, Viviane | CIRM H

Multi angle

Research School;Dynamical Systems and Ordinary Differential Equations

When a dynamical system admitting a natural (SRB) measure is perturbed, it is natural to ask how the SRB measure responds to the perturbation. In the tamest cases, this response is linear, and the derivative of the SRB measure with respect to the parameter can be expressed as a sum of decorrelations (involving the derivative of the system with respect to the parameter). In more subtle situations - for example, systems with bifurcations, or observables with singularities - the SRB measure may be a Hölder function of the parameter. This talk will present a panorama of results about linear and fractional response. When a dynamical system admitting a natural (SRB) measure is perturbed, it is natural to ask how the SRB measure responds to the perturbation. In the tamest cases, this response is linear, and the derivative of the SRB measure with respect to the parameter can be expressed as a sum of decorrelations (involving the derivative of the system with respect to the parameter). In more subtle situations - for example, systems with bifurcations, or ...

37D20

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## Intermediate dimensions, capacities and projections Falconer, Kenneth | CIRM H

Multi angle

Research talks;Analysis and its Applications

The talk will review recent work on intermediate dimensions which interpolate between Hausdorff and box dimensions. We relate these dimensions to capacities which leading to ‘Marstrand-type’ theorems on the intermediate dimensions of projections of a set in $\mathbb{R}^{n}$ onto almost all m-dimensional subspaces. This is collaborative work with various combinations of Stuart Burrell, Jonathan Fraser, Tom Kempton and Pablo Shmerkin.

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## Equilibrium measures and homoclinic classes Buzzi, Jérôme | CIRM H

Multi angle

Research School;Dynamical Systems and Ordinary Differential Equations

Works by Sarig and Benovadia have built symbolic dynamics for arbitrary diffeomorphisms of compact manifolds. This shows thatthere can be at most countably many ergodic hyperbolic equilibriummeasures for any Holder continuous or geometric potentials. We will explain how this yields uniqueness inside each homoclinic class of measures, i.e., of ergodic and hyperbolic measures that are homoclinically related. In some cases, further topological or geometric arguments can show global uniqueness.
This is a joint work with Sylvain Crovisier and Omri Sarig
Works by Sarig and Benovadia have built symbolic dynamics for arbitrary diffeomorphisms of compact manifolds. This shows thatthere can be at most countably many ergodic hyperbolic equilibriummeasures for any Holder continuous or geometric potentials. We will explain how this yields uniqueness inside each homoclinic class of measures, i.e., of ergodic and hyperbolic measures that are homoclinically related. In some cases, further topological or ...

37C40

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## Entropy and growth of periodic orbits for Anosov flows and their covers Sharp, Richard | CIRM H

Multi angle

Research School;Dynamical Systems and Ordinary Differential Equations

In this talk, we will discuss various growth rates associated to Anosov flows and their covers. The topological entropy of an Anosov flow on a compact manifold is realised as the exponential growth rate of its periodic orbits. If we pass to a regular cover of the manifold then we can consider a corresponding growth rate for the lifted flow. This growth is bounded above by the topological entropy but if the cover is infinite then the growth rate may be strictly smaller. For abelian covers, this phenomenon admits a precise description in terms of a variational principle. More recent work, joint with Rhiannon Dougall, considers more general infinite covers. In this talk, we will discuss various growth rates associated to Anosov flows and their covers. The topological entropy of an Anosov flow on a compact manifold is realised as the exponential growth rate of its periodic orbits. If we pass to a regular cover of the manifold then we can consider a corresponding growth rate for the lifted flow. This growth is bounded above by the topological entropy but if the cover is infinite then the growth rate ...

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## Closed geodesics and the measure of maximal entropy on surfaces without conjugate points Climenhaga, Vaughn | CIRM H

Multi angle

Research talks;Dynamical Systems and Ordinary Differential Equations

For negatively curved Riemannian manifolds, Margulis gave an asymptotic formula for the number of closed geodesics with length below a given threshold. I will describe joint work with Gerhard Knieper and Khadim War in which we obtain the corresponding result for surfaces without conjugate points by first proving uniqueness of the measure of maximal entropy and then following the approach of recent work by Russell Ricks, who established the asymptotic estimates in the setting of CAT(0) geodesic flows. For negatively curved Riemannian manifolds, Margulis gave an asymptotic formula for the number of closed geodesics with length below a given threshold. I will describe joint work with Gerhard Knieper and Khadim War in which we obtain the corresponding result for surfaces without conjugate points by first proving uniqueness of the measure of maximal entropy and then following the approach of recent work by Russell Ricks, who established the ...

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## Central limit theorems for circle packings Pollicott, Mark | CIRM H

Multi angle

Research talks;Dynamical Systems and Ordinary Differential Equations;Geometry;Number Theory

Given the Apollonian Circle packing, or something similar, one can consider the distribution of the logarithms of the radii. These can be shown to satisfy a Central Limit Theorem. The method of proof uses iterated function schemes and transfer operators and has applications to other conformal dynamical systems.

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## Besov spaces in multifractal environment, and the Frisch-Parisi conjecture Seuret, Stéphane | CIRM H

Multi angle

Research talks;Dynamical Systems and Ordinary Differential Equations

Multifractal properties of data coming from many scientific fields (especially in turbulence) are now rigorously established. Unfortunately, the parameters measured on these data do not correspond to those mathematically obtained for the typical (or almost sure) functions in the standard functional spaces: Hölder, Sobolev, Besov…
In this talk, we introduce very natural Besov spaces in which typical functions possess very rich scaling properties, mimicking those observed on data for instance. We obtain various characterizations of these function spaces, in terms of oscillations or wavelet coefficients.
Combining this with the construction of almost-doubling measures with prescribed scaling properties, we are able to bring a solution to the so-called Frisch-Parisi conjecture. This is a joint work with Julien Barral (Université Paris-Nord).
Multifractal properties of data coming from many scientific fields (especially in turbulence) are now rigorously established. Unfortunately, the parameters measured on these data do not correspond to those mathematically obtained for the typical (or almost sure) functions in the standard functional spaces: Hölder, Sobolev, Besov…
In this talk, we introduce very natural Besov spaces in which typical functions possess very rich scaling properties, ...

37F35

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## A brief introduction to concentration inequalities Chazottes, Jean-René | CIRM H

Multi angle

Research School;Dynamical Systems and Ordinary Differential Equations;Probability and Statistics

$Let (X,T)$ be a dynamical system preserving a probability measure $\mu$. A concentration inequality quantifies how small is the probability for $F(x,Tx,\ldots,T^{n-1}x)$ to deviate from $\int F(x,Tx,\ldots,T^{n-1}x) \mathrm{d}\mu(x)$ by an given amount $u$, where $F:X^n\to\mathbb{R}$ is supposed to be separately Lipschitz. The bound on that probability involves a constant $C$ depending only on the dynamical system (thus independent of $n$), and $\sum_{i=0}^{n-1} \mathrm{Lip}_i(F)^2$. In the best situation, the bound is $\exp(-C u^2/\sum_{i=0}^{n-1} \mathrm{Lip}_i(F)^2)$.
After explaining how to get such a bound for independent random variables, I will show how to prove it for a Gibbs measure on a shift of finite type with a Lipschitz potential, and present examples of functions $F$ to which one can apply the inequality. Finally, I will survey some results obtained for nonuniformly hyperbolic systems modeled by Young towers.
$Let (X,T)$ be a dynamical system preserving a probability measure $\mu$. A concentration inequality quantifies how small is the probability for $F(x,Tx,\ldots,T^{n-1}x)$ to deviate from $\int F(x,Tx,\ldots,T^{n-1}x) \mathrm{d}\mu(x)$ by an given amount $u$, where $F:X^n\to\mathbb{R}$ is supposed to be separately Lipschitz. The bound on that probability involves a constant $C$ depending only on the dynamical system (thus independent of $n$), ...

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## Zeta functions and the periodic orbit structure of hyperbolic dynamics Parry, William ; Pollicott, Mark | SMF 1990

Ouvrage

- 267 p.
ISBN

Astérisque , 0187

Localisation : Périodique 1er étage

équation différentielle ordinaire # fonction zêta # système dynamique

58Fxx

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## Lecture on ergotic theory and pesin theory on compact manifolds Pollicott, Mark | Cambridge University Press 1993

Ouvrage

- 162 p.
ISBN 978-0-521-43593-2

London mathematical society lecture note series , 0180

Localisation : Collection 1er étage

difféomorphisme # ensemble de Pesin # entropie # ergodicité # mesure d'invariants # point périodique # structure chaotique # théorie de la mesure # théorie ergodique # variété stable

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## Equilibrium states in negative curvature Paulin, Frédéric ; Pollicott, Mark ; Schapira, Barbara | Société Mathématique de France 2015

Ouvrage

- viii; 281 p.
ISBN 978-2-85629-818-3

Astérisque , 0373

Localisation : Périodique 1er étage

flot géodésique # courbure négative # état de Gibbs # période # dénombrement d'orbites # densité de Patterson # pression # principe variationnel # foliation instable et forte

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