New groups found with critical exponents close to but less than max.
problem Finding discrete isometry groups with critical exponents near maximum.
method Analyzing complex hyperbolic spaces to construct groups.
result Discrete isometry groups with critical exponents arbitrarily close to max but less.
New proof for certain groups in higher dimensions.
problem Properties of discrete subgroups in higher dimensions.
method Proving convex-cocompactness for specific groups.
result Finitely generated Kleinian groups with small critical exponent are convex-cocompact.
Study shows critical exponents for tree-acting groups.
problem Understanding critical exponents of discrete groups on trees.
method Explicit construction of edge-indexed graphs.
result Proven existence of groups with specific critical exponents.
Totally geodesic limit of quasi-Fuchsian groups with converging critical exponent.
problem Understanding convergence of quasi-Fuchsian groups.
method Critical exponent convergence analysis.
result Convergence of quasi-Fuchsian groups to totally geodesic representations.
Constructs free semigroups with critical exponents close to but less than ambient groups.
problem Creating free semigroups with critical exponents close to but less than ambient groups.
method Constructing finitely generated free subsemigroups with specific properties.
result Free semigroups with critical exponents arbitrarily close to but strictly less than ambient groups.
Proves critical exponent for Θ−positive representations in discrete subgroups.
problem Determining the critical exponent for Θ−positive representations. method Analyzes discrete subgroups Γ⊂PSL(2,R) and their geometric properties. result Equality of critical exponent holds if and only if Γ is a lattice for geometrically finite Γ. Study critical exponent for geodesic currents using quasi-metric spaces.
problem Understanding the critical exponent for geodesic currents.
method Associated a quasi-metric space to geodesic currents and defined a metric for filling currents, studying the critical exponent and its relation to curve intersection growth.
result The critical exponent equals the exponential growth rate of the intersection function for closed curves.
Study critical exponents in normal subgroups of higher rank Lie groups.
problem Understanding critical exponents in normal subgroups of higher rank Lie groups.
method Analyzing subgroups and their critical exponents in a higher rank semi-simple Lie group.
result Critical exponents of normal subgroups coincide under certain conditions.
Study critical exponents on hyperbolic surfaces with long boundaries using Weil-Petersson measures.
problem Analyzing critical exponents on hyperbolic surfaces with long boundaries.
method Using spine graph construction and comparing normalized Weil-Petersson and Kontsevich measures.
result Asymptotic convergence-in-mean result of normalized Weil-Petersson measures to normalized Kontsevich measures.
Study shows critical exponents of normal subgroups behave similarly to Kazhdan distances.
problem Analyzing critical exponents and Kazhdan distances in pinched Hadamard manifolds.
method Analyzing the spectrum of transfer operators associated with subshifts of finite type.
result Critical exponents of normal subgroups have the same coarse behavior as Kazhdan distances.
The paper finds free semigroups in dense subgroups of Lie groups with critical exponents arbitrarily close to the subgroup's.
problem Finding free semigroups with critical exponents arbitrarily close to a subgroup's in dense subgroups of Lie groups.
method Analyzing Zariski dense discrete subgroups of Lie groups, showing the existence of free semigroups with critical exponents arbitrarily close to the subgroup's.
result The existence of free semigroups with critical exponents arbitrarily close to the subgroup's in dense subgroups of Lie groups.
Study shows bounds on Hausdorff dimension for limit sets of projective Anosov representations.
problem Understanding Hausdorff dimensions of limit sets for projective Anosov representations.
method Proved bounds on Hausdorff dimension using critical exponents associated to highest weight and simple root.
result Hausdorff dimension of symmetric limit set is bounded by critical exponents.
The paper proves rigidity for complex Kleinian groups.
problem Characterizing hyperconvex subgroups of complex Kleinian groups.
method Analyzing critical exponents and using representations of PSL(2, C).
result Uniform lattices in PSL(2, C) are the only (d-k)-hyperconvex subgroups with a specific critical exponent.
Study critical exponents for L^p-cohomology of higher rank Lie groups and manifolds.
problem Investigate critical exponents for vanishing L^p-cohomology in higher rank Lie groups and manifolds.
method Examine SL3(R) and 5-dimensional solvable Lie groups, use spectral sequence arguments. result Discover a continuum of quasi-isometry classes of rank 2 solvable Lie groups.
Random subgroups in hyperbolic spaces have full limit sets and bounded critical exponents.
problem Understanding stationary random subgroups in hyperbolic spaces.
method Analyzing limit sets and critical exponents of random subgroups.
result Random subgroups have full limit sets and bounded critical exponents.
Study pseudo-Riemannian hyperbolic geometry, defining and proving equal critical exponent and Hausdorff dimension.
problem Understanding the geometry of limit sets in pseudo-Riemannian hyperbolic geometry.
method Introduced a class of subgroups and defined pseudo-Riemannian critical exponent and Hausdorff dimension.
result Equal and bounded critical exponent and Hausdorff dimension of limit sets.
Let Γ be a (non-elementary) convex co-compact group of isometries of a pinched Hadamard manifold X. We show that a normal subgroup Γ0 has critical exponent equal to the critical exponent of Γ if and only if Γ/Γ0 is amenable. We prove a similar result for the exponential growth rate of closed geodesics on $…
Study critical exponents of invariant subgroups in hyperbolic spaces.
problem Understanding critical exponents of invariant subgroups in hyperbolic spaces.
method Defined critical exponent δ(μ) and used a maximal ergodic theorem for hyperbolic groups.
result Critical exponent δ(μ) > d/2 in general and δ(μ) = d for divergence type subgroups.
Analytic functions on Banach spaces with Lojasiewicz exponent 1/2 are Morse-Bott.
problem Characterizing analytic functions on Banach spaces with specific gradient properties.
method Proof of converse to the Morse-Bott property for analytic functions on Banach spaces using Lojasiewicz gradient inequality and Morse Lemma.
result Analytic functions on Banach spaces with Lojasiewicz exponent 1/2 are Morse-Bott.
Modeling financial markets as gas molecules, the paper predicts phase transitions similar to water and steam.
problem Understanding the dynamics of financial markets through phase transitions.
method Developed a lattice gas model equivalent to the Ising model on a social network, analyzing critical exponents and auto-correlations.
result Financial market dynamics exhibit phase transition-like behavior, with critical exponents analogous to water and steam.
The study shows how discrete subgroups' critical exponents relate to their Zariski density in certain groups.
problem Understanding the density of discrete subgroups in semisimple Lie groups.
method Critical exponents and unitary representations.
result Discrete subgroups with critical exponents greater than a certain value are Zariski dense.
We investigated the critical dynamics on the daily Taiwan stock exchange index (TSE) from 1971 to 2005, and the 5-min intraday data from 1996 to 2005. A global persistence exponent θp was defined for non-equilibrium critical phenomena \cite{Janssen,Majumdar}, and describing dynamic behavior in an economic index \c…
On a Riemannian compact manifold, we give existence and multiplicity results for solutions of elliptic PDE by introducing isometry invariances. When the groups we used have finite orbits, we get multiplicity results for equations with the classical critical Sobolev exponent, for instance the Yamabe equation. When there…
We prove that the relative homological dimension of a Kleinian group G does not exceed 1 + the critical exponent of G. As an application of this result we show that for a geometrically finite Kleinian group G, if the topological dimension of the limit set of G equals its Hausdorff dimension, then the limit set is a rou…
New theorem on critical exponents for group actions.
problem Critical exponents of group actions.
method Proving critical exponents coincide under co-amenability condition.
result Generalizes previous results on critical exponents.
Study on p-Laplacian problems with critical exponent, focusing on existence of solutions.
problem Existence of least energy solutions for nonlinear p-Laplacian problems with critical exponent.
method Proving existence of solutions through critical point theory and variational methods.
result Significant difference in existence results between p-Laplacian and Laplacian cases.
The study proves a theorem on Riemannian manifolds for wedge products of weakly convergent differential forms.
problem Analyzing the limiting behavior of wedge products of weakly convergent differential forms on Riemannian manifolds.
method Formulating and proving compensated compactness theorems for wedge products of differential forms on closed Riemannian manifolds.
result The theorem generalizes the div-curl lemma for vectorfields and applies to critical regularity exponents.
Generalizes inequality for complete manifolds involving homology classes.
problem Volume and simplicial volume inequality for closed manifolds.
method Extends inequality to ℓ1-norm of homology classes on complete manifolds. result Inequality involving critical exponent and mass of homology classes.
Trivial solution proof for heat equation on certain manifolds.
problem Proving trivial solutions for semilinear heat equations on specific manifolds.
method Analyzing pointwise monotonicity and boundedness over time.
result Trivial solutions exist only for certain values of p.
A new flow connects manifold invariants with critical exponents.
problem Understanding invariants of non-positively curved manifolds.
method Constructing the natural flow and relating it to the critical exponent.
result Established connections between manifold invariants and critical exponents.
The study finds conditions for improved Lp norms of eigenfunctions on compact manifolds.
problem Finding necessary and sufficient conditions for improved Lp norms of eigenfunctions on compact Riemannian manifolds. method Analyzes eigenfunctions on compact Riemannian manifolds and uses properties of half-wave operators to determine conditions for improved norms.
result Conditions for improved Lpc(M) norms are necessary and sufficient for improved norms of eigenfunctions. Geodesic distance vanishes for critical Sobolev norms on diffeomorphism groups.
problem Analyzing geodesic distance in diffeomorphism groups for critical Sobolev norms.
method Combining techniques from [JM19] and [BHP18]
result Geodesic distance vanishes for Ws,n/s norms when s∈(0,1) and sp≤n. Research examines coamenable subgroups in higher rank groups.
problem Investigates coamenable normal subgroups in higher rank groups.
method Analyzes three complementary phenomena in higher rank groups.
result Growth indicators of coamenable subgroups are not preserved but the Riemannian critical exponent remains rigid.
Study shows equivalence of two methods for solving scalar curvature problem.
problem Prescribing scalar curvature of closed Riemannian manifolds.
method Subcritical approximations or negative pseudo gradient flows.
result Equivalence of both approaches with respect to zero weak limits.
The paper studies entropy and mass loss in geodesic flows on curved spaces.
problem Entropy and mass loss in geodesic flows on negatively curved manifolds.
method Ergodic theory, critical exponents of parabolic subgroups, pressure of potentials.
result Entropy is upper semicontinuous with no mass loss, but fails with mass loss due to critical exponents.
Complex hyperbolic Kleinian groups yield Stein manifolds under certain conditions.
problem Characterizing discrete groups acting on complex hyperbolic spaces.
method Proving conditions for a discrete group to yield a Stein manifold.
result If a discrete group is convex-cocompact, torsion-free, and has a critical exponent less than 2, the quotient manifold is Stein.
The paper is devoted to elaboration of a novel specific indicator based on the modified Holder exponents. This indicator has been used for forecasting critical points of financial time series and crashes of the USA stock market. The proposed approach is based on the hypothesis, which claims that before market critical …
Paper studies Hausdorff dimension of limit sets for Anosov representations.
problem Investigating the Hausdorff dimension of limit sets of Anosov representations.
method Extending the framework of hyperconvex representations and establishing a convergence property.
result Proves the Hausdorff dimension of the limit set of a hyperconvex representation equals a critical exponent.
Compact metrics found with specific curvature properties on 3D surfaces.
problem Finding compact metrics with constant curvature on 3D surfaces.
method Blow-up analysis of Yamabe equation with critical Sobolev exponents.
result Proved the compactness of conformal metrics with constant scalar curvature and boundary mean curvature.
Compact embeddings for invariant functions in metric-measure spaces.
problem Embedding functions with symmetry in metric-measure spaces.
method Analyzing H-invariant functions in compact metric-measure spaces, extending to Riemannian manifolds. result Obtained compact Sobolev embeddings for critical exponents.
We prove the existence of Veech groups having a critical exponent strictly greater than any elementary Fuchsian group (i.e. >21) but strictly smaller than any lattice (i.e. <1). More precisely, every affine covering of a primitive L-shaped Veech surface X ramified over the singularity and a non-periodic …
Study shows flash crashes in finance are self-organized criticality events.
problem Understanding and predicting anomalous price events in high-frequency finance.
method Investigated volume distributions during flash crashes and linked them to self-organized criticality.
result Volume distributions during flash crashes indicate a diverging second moment, suggesting self-organized criticality.
New separation concepts for Anosov representations help bound Thurston asymmetric metric.
problem Understanding diverging families of Anosov representations.
method Introducing separation concepts and analyzing combinatorial invariants.
result Critical exponent asymptotic to a graph invariant.
The paper examines how the Hurst exponent can reveal market regimes.
problem Determining market efficiency and behavior using time series data.
method Kurtosis analysis and Hurst exponent to identify market regimes.
result The Hurst exponent can distinguish between market regimes.
The paper explores rigidity and flexibility of isometric extensions with critical Hölder exponent.
problem The critical Hölder exponent in isometric extensions and its implications.
method Convex integration and construction of isometric extensions.
result The Hölder exponent $θ_0= rac12$ is critical, with extensions violating the tangential connection for $θ< rac12$.
Extended Otal-Peigné's Theorem to Gromov-hyperbolic spaces.
problem Prove critical exponent equals topological entropy for group actions.
method Extended Otal-Peigné's Theorem to proper, Gromov-hyperbolic spaces.
result Critical exponent equals topological entropy for line-convex spaces.
The paper confirms conjectures about Stein manifolds formed by quotients of the ball.
problem Characterizing Stein manifolds formed by quotients of the ball.
method Analyzing discrete subgroups of PU(n,1) and their quotients.
result The quotient of the ball by geometrically finite groups is Stein.
We use a straightforward variation on a recent argument of Hezari and Rivière~\cite{HR} to obtain localized Lp-estimates for all exponents larger than or equal to the critical exponent pc=n−12(n+1). We are able to this directly by just using the Lp-bounds for spectral projection operators from our …