@article{10600,
abstract = {We show that recent results on adiabatic theory for interacting gapped many-body systems on finite lattices remain valid in the thermodynamic limit. More precisely, we prove a generalized super-adiabatic theorem for the automorphism group describing the infinite volume dynamics on the quasi-local algebra of observables. The key assumption is the existence of a sequence of gapped finite volume Hamiltonians, which generates the same infinite volume dynamics in the thermodynamic limit. Our adiabatic theorem also holds for certain perturbations of gapped ground states that close the spectral gap (so it is also an adiabatic theorem for resonances and, in this sense, “generalized”), and it provides an adiabatic approximation to all orders in the adiabatic parameter (a property often called “super-adiabatic”). In addition to the existing results for finite lattices, we also perform a resummation of the adiabatic expansion and allow for observables that are not strictly local. Finally, as an application, we prove the validity of linear and higher order response theory for our class of perturbations for infinite systems. While we consider the result and its proof as new and interesting in itself, we also lay the foundation for the proof of an adiabatic theorem for systems with a gap only in the bulk, which will be presented in a follow-up article.},
author = {Henheik, Sven Joscha and Teufel, Stefan},
issn = {1089-7658},
journal = {Journal of Mathematical Physics},
keywords = {mathematical physics, statistical and nonlinear physics},
number = {1},
publisher = {AIP Publishing},
title = {{Adiabatic theorem in the thermodynamic limit: Systems with a uniform gap}},
doi = {10.1063/5.0051632},
volume = {63},
year = {2022},
}
@article{10642,
abstract = {Based on a result by Yarotsky (J Stat Phys 118, 2005), we prove that localized but otherwise arbitrary perturbations of weakly interacting quantum spin systems with uniformly gapped on-site terms change the ground state of such a system only locally, even if they close the spectral gap. We call this a strong version of the local perturbations perturb locally (LPPL) principle which is known to hold for much more general gapped systems, but only for perturbations that do not close the spectral gap of the Hamiltonian. We also extend this strong LPPL-principle to Hamiltonians that have the appropriate structure of gapped on-site terms and weak interactions only locally in some region of space. While our results are technically corollaries to a theorem of Yarotsky, we expect that the paradigm of systems with a locally gapped ground state that is completely insensitive to the form of the Hamiltonian elsewhere extends to other situations and has important physical consequences.},
author = {Henheik, Sven Joscha and Teufel, Stefan and Wessel, Tom},
issn = {1573-0530},
journal = {Letters in Mathematical Physics},
keywords = {mathematical physics, statistical and nonlinear physics},
number = {1},
publisher = {Springer Nature},
title = {{Local stability of ground states in locally gapped and weakly interacting quantum spin systems}},
doi = {10.1007/s11005-021-01494-y},
volume = {112},
year = {2022},
}
@article{10643,
abstract = {We prove a generalised super-adiabatic theorem for extended fermionic systems assuming a spectral gap only in the bulk. More precisely, we assume that the infinite system has a unique ground state and that the corresponding Gelfand–Naimark–Segal Hamiltonian has a spectral gap above its eigenvalue zero. Moreover, we show that a similar adiabatic theorem also holds in the bulk of finite systems up to errors that vanish faster than any inverse power of the system size, although the corresponding finite-volume Hamiltonians need not have a spectral gap.
},
author = {Henheik, Sven Joscha and Teufel, Stefan},
issn = {2050-5094},
journal = {Forum of Mathematics, Sigma},
keywords = {computational mathematics, discrete mathematics and combinatorics, geometry and topology, mathematical physics, statistics and probability, algebra and number theory, theoretical computer science, analysis},
publisher = {Cambridge University Press},
title = {{Adiabatic theorem in the thermodynamic limit: Systems with a gap in the bulk}},
doi = {10.1017/fms.2021.80},
volume = {10},
year = {2022},
}
@article{11783,
abstract = {We consider a gas of N bosons with interactions in the mean-field scaling regime. We review the proof of an asymptotic expansion of its low-energy spectrum, eigenstates, and dynamics, which provides corrections to Bogoliubov theory to all orders in 1/ N. This is based on joint works with Petrat, Pickl, Seiringer, and Soffer. In addition, we derive a full asymptotic expansion of the ground state one-body reduced density matrix.},
author = {Bossmann, Lea},
issn = {1089-7658},
journal = {Journal of Mathematical Physics},
keywords = {Mathematical Physics, Statistical and Nonlinear Physics},
number = {6},
publisher = {AIP Publishing},
title = {{Low-energy spectrum and dynamics of the weakly interacting Bose gas}},
doi = {10.1063/5.0089983},
volume = {63},
year = {2022},
}
@article{11732,
abstract = {We study the BCS energy gap Ξ in the high–density limit and derive an asymptotic formula, which strongly depends on the strength of the interaction potential V on the Fermi surface. In combination with the recent result by one of us (Math. Phys. Anal. Geom. 25, 3, 2022) on the critical temperature Tc at high densities, we prove the universality of the ratio of the energy gap and the critical temperature.},
author = {Henheik, Sven Joscha and Lauritsen, Asbjørn Bækgaard},
issn = {0022-4715},
journal = {Journal of Statistical Physics},
keywords = {Mathematical Physics, Statistical and Nonlinear Physics},
publisher = {Springer Nature},
title = {{The BCS energy gap at high density}},
doi = {10.1007/s10955-022-02965-9},
volume = {189},
year = {2022},
}
@article{11917,
abstract = {We study the many-body dynamics of an initially factorized bosonic wave function in the mean-field regime. We prove large deviation estimates for the fluctuations around the condensate. We derive an upper bound extending a recent result to more general interactions. Furthermore, we derive a new lower bound which agrees with the upper bound in leading order.},
author = {Rademacher, Simone Anna Elvira and Seiringer, Robert},
issn = {1572-9613},
journal = {Journal of Statistical Physics},
keywords = {Mathematical Physics, Statistical and Nonlinear Physics},
publisher = {Springer Nature},
title = {{Large deviation estimates for weakly interacting bosons}},
doi = {10.1007/s10955-022-02940-4},
volume = {188},
year = {2022},
}
@article{10623,
abstract = {We investigate the BCS critical temperature Tc in the high-density limit and derive an asymptotic formula, which strongly depends on the behavior of the interaction potential V on the Fermi-surface. Our results include a rigorous confirmation for the behavior of Tc at high densities proposed by Langmann et al. (Phys Rev Lett 122:157001, 2019) and identify precise conditions under which superconducting domes arise in BCS theory.},
author = {Henheik, Sven Joscha},
issn = {1572-9656},
journal = {Mathematical Physics, Analysis and Geometry},
keywords = {geometry and topology, mathematical physics},
number = {1},
publisher = {Springer Nature},
title = {{The BCS critical temperature at high density}},
doi = {10.1007/s11040-021-09415-0},
volume = {25},
year = {2022},
}
@article{12148,
abstract = {We prove a general local law for Wigner matrices that optimally handles observables of arbitrary rank and thus unifies the well-known averaged and isotropic local laws. As an application, we prove a central limit theorem in quantum unique ergodicity (QUE): that is, we show that the quadratic forms of a general deterministic matrix A on the bulk eigenvectors of a Wigner matrix have approximately Gaussian fluctuation. For the bulk spectrum, we thus generalise our previous result [17] as valid for test matrices A of large rank as well as the result of Benigni and Lopatto [7] as valid for specific small-rank observables.},
author = {Cipolloni, Giorgio and Erdös, László and Schröder, Dominik J},
issn = {2050-5094},
journal = {Forum of Mathematics, Sigma},
keywords = {Computational Mathematics, Discrete Mathematics and Combinatorics, Geometry and Topology, Mathematical Physics, Statistics and Probability, Algebra and Number Theory, Theoretical Computer Science, Analysis},
publisher = {Cambridge University Press},
title = {{Rank-uniform local law for Wigner matrices}},
doi = {10.1017/fms.2022.86},
volume = {10},
year = {2022},
}
@article{12145,
abstract = {In the class of strictly convex smooth boundaries each of which has no strip around its boundary foliated by invariant curves, we prove that the Taylor coefficients of the “normalized” Mather’s β-function are invariant under C∞-conjugacies. In contrast, we prove that any two elliptic billiard maps are C0-conjugate near their respective boundaries, and C∞-conjugate, near the boundary and away from a line passing through the center of the underlying ellipse. We also prove that, if the billiard maps corresponding to two ellipses are topologically conjugate, then the two ellipses are similar.},
author = {Koudjinan, Edmond and Kaloshin, Vadim},
issn = {1468-4845},
journal = {Regular and Chaotic Dynamics},
keywords = {Mechanical Engineering, Applied Mathematics, Mathematical Physics, Modeling and Simulation, Statistical and Nonlinear Physics, Mathematics (miscellaneous)},
number = {6},
pages = {525--537},
publisher = {Springer Nature},
title = {{On some invariants of Birkhoff billiards under conjugacy}},
doi = {10.1134/S1560354722050021},
volume = {27},
year = {2022},
}
@article{12232,
abstract = {We derive a precise asymptotic formula for the density of the small singular values of the real Ginibre matrix ensemble shifted by a complex parameter z as the dimension tends to infinity. For z away from the real axis the formula coincides with that for the complex Ginibre ensemble we derived earlier in Cipolloni et al. (Prob Math Phys 1:101–146, 2020). On the level of the one-point function of the low lying singular values we thus confirm the transition from real to complex Ginibre ensembles as the shift parameter z becomes genuinely complex; the analogous phenomenon has been well known for eigenvalues. We use the superbosonization formula (Littelmann et al. in Comm Math Phys 283:343–395, 2008) in a regime where the main contribution comes from a three dimensional saddle manifold.},
author = {Cipolloni, Giorgio and Erdös, László and Schröder, Dominik J},
issn = {1424-0661},
journal = {Annales Henri Poincaré},
keywords = {Mathematical Physics, Nuclear and High Energy Physics, Statistical and Nonlinear Physics},
number = {11},
pages = {3981--4002},
publisher = {Springer Nature},
title = {{Density of small singular values of the shifted real Ginibre ensemble}},
doi = {10.1007/s00023-022-01188-8},
volume = {23},
year = {2022},
}
@article{12243,
abstract = {We consider the eigenvalues of a large dimensional real or complex Ginibre matrix in the region of the complex plane where their real parts reach their maximum value. This maximum follows the Gumbel distribution and that these extreme eigenvalues form a Poisson point process as the dimension asymptotically tends to infinity. In the complex case, these facts have already been established by Bender [Probab. Theory Relat. Fields 147, 241 (2010)] and in the real case by Akemann and Phillips [J. Stat. Phys. 155, 421 (2014)] even for the more general elliptic ensemble with a sophisticated saddle point analysis. The purpose of this article is to give a very short direct proof in the Ginibre case with an effective error term. Moreover, our estimates on the correlation kernel in this regime serve as a key input for accurately locating [Formula: see text] for any large matrix X with i.i.d. entries in the companion paper [G. Cipolloni et al., arXiv:2206.04448 (2022)]. },
author = {Cipolloni, Giorgio and Erdös, László and Schröder, Dominik J and Xu, Yuanyuan},
issn = {1089-7658},
journal = {Journal of Mathematical Physics},
keywords = {Mathematical Physics, Statistical and Nonlinear Physics},
number = {10},
publisher = {AIP Publishing},
title = {{Directional extremal statistics for Ginibre eigenvalues}},
doi = {10.1063/5.0104290},
volume = {63},
year = {2022},
}
@article{12246,
abstract = {The Lieb–Oxford inequality provides a lower bound on the Coulomb energy of a classical system of N identical charges only in terms of their one-particle density. We prove here a new estimate on the best constant in this inequality. Numerical evaluation provides the value 1.58, which is a significant improvement to the previously known value 1.64. The best constant has recently been shown to be larger than 1.44. In a second part, we prove that the constant can be reduced to 1.25 when the inequality is restricted to Hartree–Fock states. This is the first proof that the exchange term is always much lower than the full indirect Coulomb energy.},
author = {Lewin, Mathieu and Lieb, Elliott H. and Seiringer, Robert},
issn = {0377-9017},
journal = {Letters in Mathematical Physics},
keywords = {Mathematical Physics, Statistical and Nonlinear Physics},
number = {5},
publisher = {Springer Nature},
title = {{Improved Lieb–Oxford bound on the indirect and exchange energies}},
doi = {10.1007/s11005-022-01584-5},
volume = {112},
year = {2022},
}
@article{12259,
abstract = {Theoretical foundations of chaos have been predominantly laid out for finite-dimensional dynamical systems, such as the three-body problem in classical mechanics and the Lorenz model in dissipative systems. In contrast, many real-world chaotic phenomena, e.g., weather, arise in systems with many (formally infinite) degrees of freedom, which limits direct quantitative analysis of such systems using chaos theory. In the present work, we demonstrate that the hydrodynamic pilot-wave systems offer a bridge between low- and high-dimensional chaotic phenomena by allowing for a systematic study of how the former connects to the latter. Specifically, we present experimental results, which show the formation of low-dimensional chaotic attractors upon destabilization of regular dynamics and a final transition to high-dimensional chaos via the merging of distinct chaotic regions through a crisis bifurcation. Moreover, we show that the post-crisis dynamics of the system can be rationalized as consecutive scatterings from the nonattracting chaotic sets with lifetimes following exponential distributions. },
author = {Choueiri, George H and Suri, Balachandra and Merrin, Jack and Serbyn, Maksym and Hof, Björn and Budanur, Nazmi B},
issn = {1089-7682},
journal = {Chaos: An Interdisciplinary Journal of Nonlinear Science},
keywords = {Applied Mathematics, General Physics and Astronomy, Mathematical Physics, Statistical and Nonlinear Physics},
number = {9},
publisher = {AIP Publishing},
title = {{Crises and chaotic scattering in hydrodynamic pilot-wave experiments}},
doi = {10.1063/5.0102904},
volume = {32},
year = {2022},
}
@article{10852,
abstract = { We review old and new results on the Fröhlich polaron model. The discussion includes the validity of the (classical) Pekar approximation in the strong coupling limit, quantum corrections to this limit, as well as the divergence of the effective polaron mass.},
author = {Seiringer, Robert},
issn = {0129-055X},
journal = {Reviews in Mathematical Physics},
keywords = {Mathematical Physics, Statistical and Nonlinear Physics},
number = {01},
publisher = {World Scientific Publishing},
title = {{The polaron at strong coupling}},
doi = {10.1142/s0129055x20600120},
volume = {33},
year = {2021},
}
@article{9121,
abstract = {We show that the energy gap for the BCS gap equation is
Ξ=μ(8e−2+o(1))exp(π2μ−−√a)
in the low density limit μ→0. Together with the similar result for the critical temperature by Hainzl and Seiringer (Lett Math Phys 84: 99–107, 2008), this shows that, in the low density limit, the ratio of the energy gap and critical temperature is a universal constant independent of the interaction potential V. The results hold for a class of potentials with negative scattering length a and no bound states.},
author = {Lauritsen, Asbjørn Bækgaard},
issn = {0377-9017},
journal = {Letters in Mathematical Physics},
keywords = {Mathematical Physics, Statistical and Nonlinear Physics},
publisher = {Springer Nature},
title = {{The BCS energy gap at low density}},
doi = {10.1007/s11005-021-01358-5},
volume = {111},
year = {2021},
}
@article{9285,
abstract = {We first review the problem of a rigorous justification of Kubo’s formula for transport coefficients in gapped extended Hamiltonian quantum systems at zero temperature. In particular, the theoretical understanding of the quantum Hall effect rests on the validity of Kubo’s formula for such systems, a connection that we review briefly as well. We then highlight an approach to linear response theory based on non-equilibrium almost-stationary states (NEASS) and on a corresponding adiabatic theorem for such systems that was recently proposed and worked out by one of us in [51] for interacting fermionic systems on finite lattices. In the second part of our paper, we show how to lift the results of [51] to infinite systems by taking a thermodynamic limit.},
author = {Henheik, Sven Joscha and Teufel, Stefan},
issn = {0129-055X},
journal = {Reviews in Mathematical Physics},
keywords = {Mathematical Physics, Statistical and Nonlinear Physics},
number = {01},
publisher = {World Scientific Publishing},
title = {{Justifying Kubo’s formula for gapped systems at zero temperature: A brief review and some new results}},
doi = {10.1142/s0129055x20600041},
volume = {33},
year = {2021},
}
@article{9891,
abstract = {Extending on ideas of Lewin, Lieb, and Seiringer [Phys. Rev. B 100, 035127 (2019)], we present a modified “floating crystal” trial state for jellium (also known as the classical homogeneous electron gas) with density equal to a characteristic function. This allows us to show that three definitions of the jellium energy coincide in dimensions d ≥ 2, thus extending the result of Cotar and Petrache [“Equality of the Jellium and uniform electron gas next-order asymptotic terms for Coulomb and Riesz potentials,” arXiv: 1707.07664 (2019)] and Lewin, Lieb, and Seiringer [Phys. Rev. B 100, 035127 (2019)] that the three definitions coincide in dimension d ≥ 3. We show that the jellium energy is also equivalent to a “renormalized energy” studied in a series of papers by Serfaty and others, and thus, by the work of Bétermin and Sandier [Constr. Approximation 47, 39–74 (2018)], we relate the jellium energy to the order n term in the logarithmic energy of n points on the unit 2-sphere. We improve upon known lower bounds for this renormalized energy. Additionally, we derive formulas for the jellium energy of periodic configurations.},
author = {Lauritsen, Asbjørn Bækgaard},
issn = {1089-7658},
journal = {Journal of Mathematical Physics},
keywords = {Mathematical Physics, Statistical and Nonlinear Physics},
number = {8},
publisher = {AIP},
title = {{Floating Wigner crystal and periodic jellium configurations}},
doi = {10.1063/5.0053494},
volume = {62},
year = {2021},
}
@article{9973,
abstract = {In this article we introduce a complete gradient estimate for symmetric quantum Markov semigroups on von Neumann algebras equipped with a normal faithful tracial state, which implies semi-convexity of the entropy with respect to the recently introduced noncommutative 2-Wasserstein distance. We show that this complete gradient estimate is stable under tensor products and free products and establish its validity for a number of examples. As an application we prove a complete modified logarithmic Sobolev inequality with optimal constant for Poisson-type semigroups on free group factors.},
author = {Wirth, Melchior and Zhang, Haonan},
issn = {1432-0916},
journal = {Communications in Mathematical Physics},
keywords = {Mathematical Physics, Statistical and Nonlinear Physics},
pages = {761–791},
publisher = {Springer Nature},
title = {{Complete gradient estimates of quantum Markov semigroups}},
doi = {10.1007/s00220-021-04199-4},
volume = {387},
year = {2021},
}
@article{8415,
abstract = {We consider billiards obtained by removing three strictly convex obstacles satisfying the non-eclipse condition on the plane. The restriction of the dynamics to the set of non-escaping orbits is conjugated to a subshift on three symbols that provides a natural labeling of all periodic orbits. We study the following inverse problem: does the Marked Length Spectrum (i.e., the set of lengths of periodic orbits together with their labeling), determine the geometry of the billiard table? We show that from the Marked Length Spectrum it is possible to recover the curvature at periodic points of period two, as well as the Lyapunov exponent of each periodic orbit.},
author = {Bálint, Péter and De Simoi, Jacopo and Kaloshin, Vadim and Leguil, Martin},
issn = {0010-3616},
journal = {Communications in Mathematical Physics},
keywords = {Mathematical Physics, Statistical and Nonlinear Physics},
number = {3},
pages = {1531--1575},
publisher = {Springer Nature},
title = {{Marked length spectrum, homoclinic orbits and the geometry of open dispersing billiards}},
doi = {10.1007/s00220-019-03448-x},
volume = {374},
year = {2019},
}
@article{8417,
abstract = {The restricted planar elliptic three body problem (RPETBP) describes the motion of a massless particle (a comet or an asteroid) under the gravitational field of two massive bodies (the primaries, say the Sun and Jupiter) revolving around their center of mass on elliptic orbits with some positive eccentricity. The aim of this paper is to show the existence of orbits whose angular momentum performs arbitrary excursions in a large region. In particular, there exist diffusive orbits, that is, with a large variation of angular momentum. The leading idea of the proof consists in analyzing parabolic motions of the comet. By a well-known result of McGehee, the union of future (resp. past) parabolic orbits is an analytic manifold P+ (resp. P−). In a properly chosen coordinate system these manifolds are stable (resp. unstable) manifolds of a manifold at infinity P∞, which we call the manifold at parabolic infinity. On P∞ it is possible to define two scattering maps, which contain the map structure of the homoclinic trajectories to it, i.e. orbits parabolic both in the future and the past. Since the inner dynamics inside P∞ is trivial, two different scattering maps are used. The combination of these two scattering maps permits the design of the desired diffusive pseudo-orbits. Using shadowing techniques and these pseudo orbits we show the existence of true trajectories of the RPETBP whose angular momentum varies in any predetermined fashion.},
author = {Delshams, Amadeu and Kaloshin, Vadim and de la Rosa, Abraham and Seara, Tere M.},
issn = {0010-3616},
journal = {Communications in Mathematical Physics},
keywords = {Mathematical Physics, Statistical and Nonlinear Physics},
number = {3},
pages = {1173--1228},
publisher = {Springer Nature},
title = {{Global instability in the restricted planar elliptic three body problem}},
doi = {10.1007/s00220-018-3248-z},
volume = {366},
year = {2018},
}
@article{8420,
abstract = {We show that in the space of all convex billiard boundaries, the set of boundaries with rational caustics is dense. More precisely, the set of billiard boundaries with caustics of rotation number 1/q is polynomially sense in the smooth case, and exponentially dense in the analytic case.},
author = {Kaloshin, Vadim and Zhang, Ke},
issn = {0951-7715},
journal = {Nonlinearity},
keywords = {Mathematical Physics, General Physics and Astronomy, Applied Mathematics, Statistical and Nonlinear Physics},
number = {11},
pages = {5214--5234},
publisher = {IOP Publishing},
title = {{Density of convex billiards with rational caustics}},
doi = {10.1088/1361-6544/aadc12},
volume = {31},
year = {2018},
}
@article{8498,
abstract = {In the present note we announce a proof of a strong form of Arnold diffusion for smooth convex Hamiltonian systems. Let ${\mathbb T}^2$ be a 2-dimensional torus and B2 be the unit ball around the origin in ${\mathbb R}^2$ . Fix ρ > 0. Our main result says that for a 'generic' time-periodic perturbation of an integrable system of two degrees of freedom $H_0(p)+\varepsilon H_1(\theta,p,t),\quad \ \theta\in {\mathbb T}^2,\ p\in B^2,\ t\in {\mathbb T}={\mathbb R}/{\mathbb Z}$ , with a strictly convex H0, there exists a ρ-dense orbit (θε, pε, t)(t) in ${\mathbb T}^2 \times B^2 \times {\mathbb T}$ , namely, a ρ-neighborhood of the orbit contains ${\mathbb T}^2 \times B^2 \times {\mathbb T}$ .
Our proof is a combination of geometric and variational methods. The fundamental elements of the construction are the usage of crumpled normally hyperbolic invariant cylinders from [9], flower and simple normally hyperbolic invariant manifolds from [36] as well as their kissing property at a strong double resonance. This allows us to build a 'connected' net of three-dimensional normally hyperbolic invariant manifolds. To construct diffusing orbits along this net we employ a version of the Mather variational method [41] equipped with weak KAM theory [28], proposed by Bernard in [7].},
author = {Kaloshin, Vadim and Zhang, K},
issn = {0951-7715},
journal = {Nonlinearity},
keywords = {Mathematical Physics, General Physics and Astronomy, Applied Mathematics, Statistical and Nonlinear Physics},
number = {8},
pages = {2699--2720},
publisher = {IOP Publishing},
title = {{Arnold diffusion for smooth convex systems of two and a half degrees of freedom}},
doi = {10.1088/0951-7715/28/8/2699},
volume = {28},
year = {2015},
}
@article{8502,
abstract = {The famous ergodic hypothesis suggests that for a typical Hamiltonian on a typical energy surface nearly all trajectories are dense. KAM theory disproves it. Ehrenfest (The Conceptual Foundations of the Statistical Approach in Mechanics. Ithaca, NY: Cornell University Press, 1959) and Birkhoff (Collected Math Papers. Vol 2, New York: Dover, pp 462–465, 1968) stated the quasi-ergodic hypothesis claiming that a typical Hamiltonian on a typical energy surface has a dense orbit. This question is wide open. Herman (Proceedings of the International Congress of Mathematicians, Vol II (Berlin, 1998). Doc Math 1998, Extra Vol II, Berlin: Int Math Union, pp 797–808, 1998) proposed to look for an example of a Hamiltonian near H0(I)=⟨I,I⟩2 with a dense orbit on the unit energy surface. In this paper we construct a Hamiltonian H0(I)+εH1(θ,I,ε) which has an orbit dense in a set of maximal Hausdorff dimension equal to 5 on the unit energy surface.},
author = {Kaloshin, Vadim and Saprykina, Maria},
issn = {0010-3616},
journal = {Communications in Mathematical Physics},
keywords = {Mathematical Physics, Statistical and Nonlinear Physics},
number = {3},
pages = {643--697},
publisher = {Springer Nature},
title = {{An example of a nearly integrable Hamiltonian system with a trajectory dense in a set of maximal Hausdorff dimension}},
doi = {10.1007/s00220-012-1532-x},
volume = {315},
year = {2012},
}
@article{8525,
abstract = {Let M be a smooth compact manifold of dimension at least 2 and Diffr(M) be the space of C r smooth diffeomorphisms of M. Associate to each diffeomorphism f;isin; Diffr(M) the sequence P n (f) of the number of isolated periodic points for f of period n. In this paper we exhibit an open set N in the space of diffeomorphisms Diffr(M) such for a Baire generic diffeomorphism f∈N the number of periodic points P n f grows with a period n faster than any following sequence of numbers {a n } n ∈ Z + along a subsequence, i.e. P n (f)>a ni for some n i →∞ with i→∞. In the cases of surface diffeomorphisms, i.e. dim M≡2, an open set N with a supergrowth of the number of periodic points is a Newhouse domain. A proof of the man result is based on the Gontchenko–Shilnikov–Turaev Theorem [GST]. A complete proof of that theorem is also presented.},
author = {Kaloshin, Vadim},
issn = {0010-3616},
journal = {Communications in Mathematical Physics},
keywords = {Mathematical Physics, Statistical and Nonlinear Physics},
pages = {253--271},
publisher = {Springer Nature},
title = {{Generic diffeomorphisms with superexponential growth of number of periodic orbits}},
doi = {10.1007/s002200050811},
volume = {211},
year = {2000},
}
@article{8527,
abstract = {We introduce a new potential-theoretic definition of the dimension spectrum of a probability measure for q > 1 and explain its relation to prior definitions. We apply this definition to prove that if and is a Borel probability measure with compact support in , then under almost every linear transformation from to , the q-dimension of the image of is ; in particular, the q-dimension of is preserved provided . We also present results on the preservation of information dimension and pointwise dimension. Finally, for and q > 2 we give examples for which is not preserved by any linear transformation into . All results for typical linear transformations are also proved for typical (in the sense of prevalence) continuously differentiable functions.},
author = {Hunt, Brian R and Kaloshin, Vadim},
issn = {0951-7715},
journal = {Nonlinearity},
keywords = {Mathematical Physics, General Physics and Astronomy, Applied Mathematics, Statistical and Nonlinear Physics},
number = {5},
pages = {1031--1046},
publisher = {IOP Publishing},
title = {{How projections affect the dimension spectrum of fractal measures}},
doi = {10.1088/0951-7715/10/5/002},
volume = {10},
year = {1997},
}