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Mark Demers

Publications and source records attributed to Mark Demers.

16 recordsLinked to original sources
arXiv2023

A trichotomy for hitting times and escape rates for a class of unimodal maps

We consider local escape rates and hitting time statistics for unimodal interval maps of Misiurewicz-Thurston type. We prove that for any point $z$ in the interval there is a local escape rate and hitting time statistics which is one of three types. While it is key that we cover all points $z$, the particular interest here is when $z$ is periodic and in the postcritical orbit which yields the third part of the trichotomy. We also prove generalised asymptotic escape rates of the form first shown by Bruin, Demers and Todd.

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arXiv2023

Lyapunov exponents and nonadapted measures for dispersing billiards

For hyperbolic systems with singularities, such as dispersing billiards, Pesin theory as developed by Katok and Strelcyn applies to measures that are "adapted" in the sense that they do not give too much weight to neighborhoods of the singularity set. The zero-entropy measures supported on grazing periodic orbits are nonadapted, but it has been an open question whether there are nonadapted measures with positive entropy. We construct such measures for any dispersing billiard with a periodic orbit having a single grazing collision; we then use our construction to show that the thermodynamic formalism for such billiards has a phase transition even when one restricts attention to adapted or to positive entropy measures.

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arXiv2022

Measure of maximal entropy for finite horizon Sinai billiard flows

Using recent work of Carrand on equilibrium states for the billiard map, and bootstrapping via a "leapfrogging" method from a previous article of Baladi and Demers, we construct the unique measure of maximal entropy for two-dimensional finite horizon Sinai (dispersive) billiard flows (and show it is Bernoulli), assuming that the topological entropy of the flow is strictly larger than s_0 log 2 where 0<s_0<1 quantifies the recurrence to singularities. This bound holds in many examples (it is expected to hold generically).

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arXiv2020

Thermodynamic formalism for dispersing billiards

For any finite horizon Sinai billiard map T on the two-torus, we find t_*>1 such that for each t in (0,t_*) there exists a unique equilibrium state $\mu_t$ for $- t\log J^uT$, and $\mu_t$ is T-adapted. (In particular, the SRB measure is the unique equilibrium state for $- \log J^uT$.) We show that $\mu_t$ is exponentially mixing for Holder observables, and the pressure function $P(t)=\sup_\mu \{h_\mu -\int t\log J^uT d \mu\}$ is analytic on (0,t_*). In addition, P(t) is strictly convex if and only if $\log J^uT$ is not $\mu_t$ a.e. cohomologous to a constant, while, if there exist $t_a\ne t_b$ with $\mu_{t_a}= \mu_{t_b}$, then P(t) is affine on (0,t_*). An additional sparse recurrence condition gives $\lim_{t\to 0} P(t)=P(0)$.

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arXiv2019

Asymptotic escape rates and limiting distributions for multimodal maps

We consider multimodal maps with holes and study the evolution of the open systems with respect to equilibrium states for both geometric and H\"older potentials. For small holes, we show that a large class of initial distributions share the same escape rate and converge to a unique absolutely continuous conditionally invariant measure; we also prove a variational principle connecting the escape rate to the pressure on the survivor set, with no conditions on the placement of the hole. Finally, introducing a weak condition on the centre of the hole, we prove scaling limits for the escape rate for holes centred at both periodic and nonperiodic points, as the diameter of the hole goes to zero.

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arXiv2018

On the Measure of Maximal Entropy for Finite Horizon Sinai Billiard Maps

The Sinai billiard map $T$ on the two-torus, i.e., the periodic Lorentz gas, is a discontinuous map. Assuming finite horizon, we propose a definition $h_*$ for the topological entropy of $T$. We prove that $h_*$ is not smaller than the value given by the variational principle, and that it is equal to the definitions of Bowen using spanning or separating sets. Under a mild condition of sparse recurrence to the singularities, we get more: First, using a transfer operator acting on a space of anisotropic distributions, we construct an invariant probability measure $\mu_*$ of maximal entropy for $T$ (i.e., $h_{\mu_*}(T)=h_*$), we show that $\mu_*$ has full support and is Bernoulli, and we prove that $\mu_*$ is the unique measure of maximal entropy, and that it is different from the smooth invariant measure except if all non grazing periodic orbits have multiplier equal to $h_*$. Second, $h_*$ is equal to the Bowen--Pesin--Pitskel topological entropy of the restriction of $T$ to a non-compact domain of continuity. Last, applying results of Lima and Matheus, as upgraded by Buzzi, the map $T$ has at least $C e^{nh_*}$ periodic points of period $n$ for all large enough $n \in \mathbb{N}$.

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arXiv2017

Fluctuation of the entropy production for the Lorentz gas under small external forces

In this paper we study the physical and statistical properties of the periodic Lorentz gas with finite horizon driven to a non-equilibrium steady state by the combination of non-conservative external forces and deterministic thermostats. A version of this model was introduced by Chernov, Eyink, Lebowitz, and Sinai and subsequently generalized by Chernov and the third author. Non-equilibrium steady states for these models are SRB measures and they are characterized by the positivity of the steady state entropy production rate. Our main result is to establish that the entropy production, in this context equal to the phase space contraction, satisfies the Gallavotti-Cohen fluctuation relation. The main tool needed in the proof is the family of anisotropic Banach spaces introduced by the first and third authors to study the ergodic and statistical properties of billiards using transfer operator techniques.

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arXiv2016

Equilibrium states, pressure and escape for multimodal maps with holes

For a class of non-uniformly hyperbolic interval maps, we study rates of escape with respect to conformal measures associated with a family of geometric potentials. We establish the existence of physically relevant conditionally invariant measures and equilibrium states and prove a relation between the rate of escape and pressure with respect to these potentials. As a consequence, we obtain a Bowen formula: we express the Hausdorff dimension of the set of points which never exit through the hole in terms of the relevant pressure function. Finally, we obtain an expression for the derivative of the escape rate in the zero-hole limit.

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arXiv2015

Exponential Decay of Correlations for Finite Horizon Sinai Billiard Flows

We prove exponential decay of correlations for the billiard flow associated with a two-dimensional finite horizon Lorentz Gas (i.e., the Sinai billiard flow with finite horizon). Along the way, we describe the spectrum of the generator of the corresponding semi-group L_t of transfer operators, i.e., the resonances of the Sinai billiard flow, on a suitable Banach space of anisotropic distributions. (Revised after referees' comments.)

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arXiv2014

Escape Rates and Singular Limiting Distributions for Intermittent Maps with Holes

We study the escape dynamics in the presence of a hole of a standard family of intermittent maps of the unit interval with neutral fixed point at the origin (and finite absolutely continuous invariant measure). Provided that the hole (is a cylinder that) does not contain any neighborhood of the origin, the surviving volume is shown to decay at polynomial speed with time. The associated polynomial escape rate depends on the density of the initial distribution, more precisely, on its behavior in the vicinity of the origin. Moreover, the associated normalized push forward measures are proved to converge to the point mass supported at the origin, in sharp contrast to systems with exponential escape rate. Finally, a similar result is obtained for more general systems with subexponential escape rates; namely that the Cesàro limit of normalized push forward measures is typically singular, invariant and supported on the asymptotic survivor set.

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arXiv2011

Behavior of the Escape Rate Function in Hyperbolic Dynamical Systems

For a fixed initial reference measure, we study the dependence of the escape rate on the hole for a smooth or piecewise smooth hyperbolic map. First, we prove the existence and Holder continuity of the escape rate for systems with small holes admitting Young towers. Then we consider general holes for Anosov diffeomorphisms, without size or Markovian restrictions. We prove bounds on the upper and lower escape rates using the notion of pressure on the survivor set and show that a variational principle holds under generic conditions. However, we also show that the escape rate function forms a devil's staircase with jumps along sequences of regular holes and present examples to elucidate some of the difficulties involved in formulating a general theory.

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arXiv2011

Entropy, Lyapunov Exponents and Escape Rates in Open Systems

We study the relation between escape rates and pressure in general dynamical systems with holes, where pressure is defined to be the difference between entropy and the sum of positive Lyapunov exponents. Central to the discussion is the formulation of a class of invariant measures supported on the survivor set over which we take the supremum to measure the pressure. Upper bounds for escape rates are proved for general diffeomorphisms of manifolds, possibly with singularities, for arbitrary holes and natural initial distributions including Lebesgue and SRB measures. Lower bounds do not hold in such generality, but for systems admitting Markov tower extensions with spectral gaps, we prove the equality of the escape rate with the absolute value of the pressure and the existence of an invariant measure realizing the escape rate, i.e. we prove a full variational principle. As an application of our results, we prove a variational principle for the billiard map associated with a planar Lorentz gas of finite horizon with holes.

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arXiv2009

Escape Rates and Physically Relevant Measures for Billiards with Small Holes

We study the billiard map corresponding to a periodic Lorentz gas in 2-dimensions in the presence of small holes in the table. We allow holes in the form of open sets away from the scatterers as well as segments on the boundaries of the scatterers. For a large class of smooth initial distributions, we establish the existence of a common escape rate and normalized limiting distribution. This limiting distribution is conditionally invariant and is the natural analogue of the SRB measure of a closed system. Finally, we prove that as the size of the hole tends to zero, the limiting distribution converges to the smooth invariant measure of the billiard map.

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arXiv2008

Existence and convergence properties of physical measures for certain dynamical systems with holes

We study two classes of dynamical systems with holes: expanding maps of the interval and Collet-Eckmann maps with singularities. In both cases, we prove that there is a natural absolutely continuous conditionally invariant measure $μ$ (a.c.c.i.m.) with the physical property that strictly positive Hölder continuous functions converge to the density of $μ$ under the renormalized dynamics of the system. In addition, we construct an invariant measure $ν$, supported on the Cantor set of points that never escape from the system, that is ergodic and enjoys exponential decay of correlations for Hölder observables. We show that $ν$ satisfies an equilibrium principle which implies that the escape rate formula, familiar to the thermodynamic formalism, holds outside the usual setting. In particular, it holds for Collet-Eckmann maps with holes, which are not uniformly hyperbolic and do not admit a finite Markov partition. We use a general framework of Young towers with holes and first prove results about the \accim and the invariant measure on the tower. Then we show how to transfer results to the original dynamical system. This approach can be expected to generalize to other dynamical systems than the two above classes.

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arXiv2006

Markov Extensions for Dynamical Systems with Holes: An Application to Expanding Maps of the Interval

We introduce the Markov extension, represented schematically as a tower, to the study of dynamical systems with holes. For tower maps with small holes, we prove the existence of conditionally invariant probability measures which are absolutely continuous with respect to Lebesgue measure (abbreviated a.c.c.i.m.). We develop restrictions on the Lebesgue measure of the holes and simple conditions on the dynamics of the tower which ensure existence and uniqueness in a class of Holder continuous densities. We then use these results to study the existence and properties of a.c.c.i.m. for expanding maps of the interval with holes. We obtain the convergence of the a.c.c.i.m. to the SRB measure of the corresponding closed system as the measure of the hole shrinks to zero.

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