Equilibrium States of Weakly Hyperbolic One-Dimensional Maps for Hölder Potentials

There is a wealth of results in the literature on the thermodynamic formalism for potentials that are, in some sense, “hyperbolic”. We show that for a sufficiently regular one-dimensional map satisfying a weak hyperbolicity assumption, every Hölder continuous potential is hyperbolic. A sample consequence is the absence of phase transitions: The pressure function is real analytic on the space of Hölder continuous functions. Another consequence is that every Hölder continuous potential has a unique equilibrium state, and that this measure has exponential decay of correlations.     

Cauchy Problem for Dissipative Hölder Solutions to the Incompressible Euler Equations

We consider solutions to the Cauchy problem for the incompressible Euler equations on the 3-dimensional torus which are continuous or Hölder continuous for any exponent ${\theta < \frac{1}{16}}$ . Using the techniques introduced in De Lellis and Székelyhidi (Inventiones Mathematicae 9:377–407, 2013; Dissipative Euler flows and Onsager’s conjecture, 2012), we prove the existence of infinitely many (Hölder) continuous initial vector fields starting from which there exist infinitely many (Hölder) continuous solutions with preassigned total kinetic energy.     

Non-lacunary Gibbs Measures for Certain Fractal Repellers

In this paper, we study non-uniformly expanding repellers constructed as the limit sets for a non-uniformly expanding dynamical systems. We prove that given a Hölder continuous potential φ satisfying a summability condition, there exists non-lacunary Gibbs measure for φ, with positive Lyapunov exponents and infinitely many hyperbolic times almost everywhere. Moreover, this non-lacunary Gibbs measure is an equilibrium measure for φ.     

Large Deviations,Fluctuations and Shrinking Intervals

This paper concerns the statistical properties of hyperbolic diffeomorphisms. We obtain a large deviation result with respect to slowly shrinking intervals for a large class of Hölder continuous functions. In case of time reversal symmetry, we obtain a corresponding version of the Fluctuation Theorem.     

Correlation Decay and Recurrence Asymptotics for Some Robust Nonuniformly Hyperbolic Maps

We study a robust class of multidimensional non-uniformly hyperbolic transformations considered by Oliveira and Viana (Ergod. Theory Dyn. Syst. 28:501–533, 2008). For an open class of Hölder continuous potentials with small variation we show that the unique equilibrium state has exponential decay of correlations and that the distribution of hitting times is asymptotically exponential. Furthermore, using that the equilibrium states satisfy a weak Gibbs property we also prove log-normal fluctuations of the return times around their average.     

Projections of Gibbs States for Hölder Potentials

We give a short proof that the projection of a Gibbs state for a Hölder continuous potential on a mixing shift of finite type under a 1-block fiber-wise mixing factor map has a Hölder continuous g function. This improves a number of previous results. The key insight in the proof is to realize the measure of a cylinder set in terms of positive operators and use cone techniques.     

On the dimension of deterministic and random Cantor-like sets,symbolic dynamics,and the Eckmann-Ruelle Conjecture

In this paper we unify and extend many of the known results on the dimension of deterministic and random Cantor-like sets in ? ⁿ , and apply these results to study some problems in dynamical systems. In particular, we verify the Eckmann-Ruelle Conjecture for equilibrium measures for Hölder continuous conformal expanding maps and conformal Axiom A^# (topologically hyperbolic) homeomorphims. We also construct a Hölder continuous Axiom A^# homeomorphism of positive topological entropy for which the unique measure of maximal entropy is ergodic and has different upper and lower pointwise dimensions almost everywhere. this example shows that the non-conformal Hölder continuous version of the Eckmann-Ruelle Conjecture is false. The Cantor-like sets we consider are defined by geometric constructions of different types. The vast majority of geometric constructions studied in the literature are generated by a finite collection ofp maps which are either contractions or similarities and are modeled by the full shift onp symbols (or at most a subshift of finite type). In this paper we consider much more general classes of geometric constructions: the placement of the basic sets at each step of the construction can be arbitrary, and they need not be disjoint. Moreover, our constructions are modeled by arbitrary symbolic dynamical systems. The importance of this is to reveal the close and nontrivial relations between the statistical mechanics (and especially the absence of phase transitions) of the symbolic dynamical system underlying the geometric construction and the dimension of its limit set. This has not been previously observed since no phase transitions can occur for subshifts of finite type. We also consider nonstationary constructions, random constructions (determined by an arbitrary ergodic stationary distribution), and combinations of the above.     

Generalized Hölder Continuity and Oscillation Functions

We study a notion of generalized Hölder continuity for functions on ?^d. We show that for any bounded function f of bounded support and any r >?0, the r-oscillation of f defined as \(osc_{r} f (x):= \sup _{B_{r}(x)} f - \inf _{B_{r}(x)} f\) is automatically generalized Hölder continuous, and we give an estimate for the appropriate (semi)norm. This is motivated by applications in the theory of dynamical systems.     

A Matrix-Valued Point Interactions Model

We study a matrix-valued Schrödinger operator with random point interactions. We prove the absence of absolutely continuous spectrum for this operator by proving that, away from a discrete set, its Lyapunov exponents do not vanish. For this we use a criterion by Gol’dsheid and Margulis and we prove the Zariski denseness, in the symplectic group, of the group generated by the transfer matrices. Then we prove estimates on the transfer matrices which lead to the Hölder continuity of the Lyapunov exponents. After proving the existence of the integrated density of states of the operator, we also prove its Hölder continuity by proving a Thouless formula which links the integrated density of states to the sum of the positive Lyapunov exponents.     

Generalized Eigenvalue-Counting Estimates for the Anderson Model

We generalize Minami’s estimate for the Anderson model and its extensions to n eigenvalues, allowing for n arbitrary intervals and arbitrary single-site probability measures with no atoms. As an application, we derive new results about the multiplicity of eigenvalues and Mott’s formula for the ac-conductivity when the single site probability distribution is Hölder continuous.     

Bounds on the Spectral Shift Function and the Density of States

We study spectra of Schrödinger operators on ? ^d . First we consider a pair of operators which differ by a compactly supported potential, as well as the corresponding semigroups. We prove almost exponential decay of the singular values μ _n of the difference of the semigroups as n→∞ and deduce bounds on the spectral shift function of the pair of operators. Thereafter we consider alloy type random Schrödinger operators. The single site potential u is assumed to be non-negative and of compact support. The distributions of the random coupling constants are assumed to be Hölder continuous. Based on the estimates for the spectral shift function, we prove a Wegner estimate which implies Hölder continuity of the integrated density of states.     

Spectral Gap and Transience for Ruelle Operators on Countable Markov Shifts

We find a necessary and sufficient condition for the Ruelle operator of a weakly Hölder continuous potential on a topologically mixing countable Markov shift to act with spectral gap on some rich Banach space. We show that the set of potentials satisfying this condition is open and dense for a variety of topologies. We then analyze the complement of this set (in a finer topology) and show that among the three known obstructions to spectral gap (weak positive recurrence, null recurrence, transience), transience is open and dense, and null recurrence and weak positive recurrence have empty interior.     

A Stretched Exponential Bound on Time Correlations for Billiard Flows

We construct Markov approximations to the billiard flows and establish a stretched exponential bound on time-correlation functions for planar periodic Lorentz gases (also known as Sinai billiards). Precisely, we show that for any (generalized) Hölder continuous functions F,G on the phase space of the flow the time correlation function is bounded by const? \(e^{-a\sqrt{|t|}}\), here t ? ? is the (continuous) time and a > 0.     

Onsager’s Conjecture Almost Everywhere in Time

Communications in Mathematical Physics - In recent works by Isett (Hölder continuous Euler flows in three dimensions with compact support in time, pp 1–173, 2012), and later by...     

Hölder Continuity of Absolutely Continuous Spectral Measures for One-Frequency Schrödinger Operators

We establish sharp results on the modulus of continuity of the distribution of the spectral measure for one-frequency Schrödinger operators with Diophantine frequencies in the region of absolutely continuous spectrum. More precisely, we establish 1/2-Hölder continuity near almost reducible energies (an essential support of absolutely continuous spectrum). For non-perturbatively small potentials (and for the almost Mathieu operator with subcritical coupling), our results apply for all energies.     

A Nonlinear Transfer Operator Theorem

In recent papers, Kenyon et al. (Ergod Theory Dyn Syst 32:1567–1584 2012), and Fan et al. (C R Math Acad Sci Paris 349:961–964 2011, Adv Math 295:271–333 2016) introduced a form of non-linear thermodynamic formalism based on solutions to a non-linear equation using matrices. In this note we consider the more general setting of Hölder continuous functions.     

On the Zero-Temperature Limit of Gibbs States

We exhibit Lipschitz (and hence Hölder) potentials on the full shift ${\{0,1\}^{\mathbb{N}}}We exhibit Lipschitz (and hence H?lder) potentials on the full shift {0,1}^\mathbbN{\{0,1\}^{\mathbb{N}}} such that the associated Gibbs measures fail to converge as the temperature goes to zero. Thus there are “exponentially decaying” interactions on the configuration space {0,1}^{\mathbb Z}{\{0,1\}^{\mathbb Z}} for which the zero-temperature limit of the associated Gibbs measures does not exist. In higher dimension, namely on the configuration space {0,1}^\mathbbZd{\{0,1\}^{\mathbb{Z}^{d}}}, d ≥ 3, we show that this non-convergence behavior can occur for the equilibrium states of finite-range interactions, that is, for locally constant potentials.     

Equilibrium States for Interval Maps: Potentials with sup φ − inf φ < h top (f)

We study an inducing scheme approach for smooth interval maps to prove existence and uniqueness of equilibrium states for potentials φ with the ‘bounded range’ condition sup φ ? inf φ < h _top (f), first used by Hofbauer and Keller [HK]. We compare our results to Hofbauer and Keller’s use of Perron-Frobenius operators. We demonstrate that this ‘bounded range’ condition on the potential is important even if the potential is Hölder continuous. We also prove analyticity of the pressure in this context.     

Continuity of the Lyapunov Exponents for Quasiperiodic Cocycles

Consider the Banach manifold of real analytic linear cocycles with values in the general linear group of any dimension and base dynamics given by a Diophantine translation on the circle. We prove a precise higher dimensional Avalanche Principle and use it in an inductive scheme to show that the Lyapunov spectrum blocks associated to a gap pattern in the Lyapunov spectrum of such a cocycle are locally Hölder continuous. Moreover, we show that all Lyapunov exponents are continuous everywhere in this Banach manifold, irrespective of any gap pattern in their spectra. These results also hold for Diophantine translations on higher dimensional tori, albeit with a loss in the modulus of continuity of the Lyapunov spectrum blocks.     

A Strong Pair Correlation Bound Implies the CLT for Sinai Billiards

We investigate the possibility of proving the Central Limit Theorem (CLT) for Dynamical Systems using only information on pair correlations. A strong bound on multiple correlations is known to imply the CLT (Chernov and Markarian in Chaotic Billiards, 2006). In Chernov’s paper (J. Stat. Phys. 122(6), 2006), such a bound is derived for dynamically Hölder continuous observables of dispersing Billiards. Here we weaken the regularity assumption and subsequently show that the bound on multiple correlations follows directly from the bound on pair correlations. Thus, a strong bound on pair correlations alone implies the CLT, for a wider class of observables. The result is extended to Anosov diffeomorphisms in any dimension. Some non-invertible maps are also considered.