arXiv subjects

Abdoulaye Thiam

Publications and source records attributed to Abdoulaye Thiam.

9 recordsLinked to original sources

arXiv2026

Transfer Operators and SRB Measures for Axiom A Diffeomorphisms: Spectral Gap, Structural Stability, and the Gibbs Equivalence Theorem

We develop the Ruelle transfer operator theory for Axiom A diffeomorphisms and construct Sinai-Ruelle-Bowen measures, carrying the symbolic spectral results of Part I [64] over to smooth dynamics through the Markov partition coding of Part III [66]. This Part contains four Main Theorems. The first proves structural stability of Axiom A diffeomorphisms satisfying the strong transversality condition, with an explicit Hölder exponent for the conjugating homeomorphism in terms of the hyperbolicity data, refining the classical results of Robbin and Robinson. The second establishes quasi-compactness of the transfer operator on Hölder spaces with a quantitative spectral gap bound; as consequences we obtain exponential decay of correlations with explicit rate, the central limit theorem for Hölder observables via the Nagaev-Guivarc'h spectral perturbation method, real-analyticity of the pressure, and meromorphic continuation of the Ruelle dynamical zeta function. The third constructs SRB measures on mixing basic sets as the unique equilibrium states for the geometric potential, proves absolute continuity of the unstable foliation, and derives an explicit product-formula for the conditional densities along unstable manifolds. The fourth establishes the Pesin entropy formula identifying the Kolmogorov-Sinai entropy of the SRB measure with the sum of its positive Lyapunov exponents. The Gibbs Equivalence Theorem, assembling the symbolic, variational, spectral, and geometric characterizations of the equilibrium state on a mixing basic set, follows from these four Main Theorems together with the imported results of Parts I and III [64,66]. This Part constitutes Part IV of a six-part series on the thermodynamic formalism for hyperbolic dynamical systems.

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arXiv2026

Statistical Limit Theorems for Axiom A Diffeomorphisms: Exponential Mixing, Central Limit Theorem, and Large Deviations

We establish statistical limit theorems for equilibrium states of Axiom A diffeomorphisms, derived from the spectral gap of the Ruelle transfer operator established in Part I (Thiam2026a) and transferred to smooth dynamics through the Markov partition coding of Part III (Thiam2026c). This Part contains five Main Theorems. The first proves the Volume Lemma with explicit two-sided bounds on the Riemannian volume of dynamical Bowen balls in terms of Birkhoff sums of the geometric potential. The second establishes exponential decay of correlations with explicit mixing rates computed from the spectral gap of the normalized transfer operator. The third proves the Central Limit Theorem with Berry-Esseen bounds at the optimal rate, with an explicit spectral formula for the asymptotic variance and a characterization of its degeneracy through the Livšic coboundary condition. The fourth establishes the Almost Sure Invariance Principle, providing pathwise Brownian approximation with polynomial error via the martingale embedding method. The fifth proves a large deviations principle with rate function given by the Legendre transform of the pressure. The individual results are due to Sinai, Ruelle, Ratner, Denker-Philipp, Gouëzel, Kifer, Melbourne-Nicol, and Young; the contribution is their derivation from a single spectral mechanism with explicit dependence on hyperbolicity data. This Part constitutes Part V of a six-part series on the thermodynamic formalism for hyperbolic dynamical systems.

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arXiv2026

Mixed Global Dynamics of the Forced Vibro-Impact Oscillator with Coulomb Friction and its Symplectic Structure, KAM Tori, and Persistence

The forced vibro-impact oscillator with Amonton-Coulomb friction and elastic walls was shown by Gendelman et al. (2019) to exhibit a coexistence of Hamiltonian stability islands and dissipative attractors in a single phase space. We provide a complete mathematical analysis of this phenomenon. We prove global well-posedness of the associated Filippov flow and construct a global lift to a piecewise smooth Hamiltonian system on a covering manifold. On the maximal forward-invariant non-sticking set, we show that the time-$T$ stroboscopic map is exact symplectic, within the formalism of symplectic dynamics. We derive a closed-form existence equation for symmetric $T$-periodic orbits and establish a parameter-dependent saddle-center bifurcation at $f_{\rm sc}(F,ω,R)$, correcting a universality claim in prior work. Using Moser's twist theorem, we prove the existence of invariant Cantor families (KAM tori) near elliptic non-sticking periodic orbits, while a Melnikov analysis yields hyperbolic dynamics conjugate to a Bernoulli shift near the associated saddle. We further show that any positive restitution defect or viscous damping destroys the conservative structure: elliptic periodic orbits persist but become asymptotically stable, replacing Hamiltonian islands by a single attracting basin. The approach extends to multi-particle systems with elastic collisions, where a symplectic structure and higher-dimensional KAM tori are obtained. A computer-assisted proof verifies the existence and ellipticity of a non-sticking periodic orbit at a specific parameter point.

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arXiv2026

Quantitative Hölder Regularity, Concentration, and Spectral Applications for Lyapunov Exponents of Random $\operatorname{GL}(2,\mathbb{R})$ Cocycles, with Extensions to $\operatorname{GL}(d,\mathbb{R})$

This paper develops a quantitative regularity theory for the Lyapunov exponents of random products of matrices in $\operatorname{GL}(2,\mathbb{R})$, with extensions to $\operatorname{GL}(d,\mathbb{R})$ for all $d \geq 2$. At every compactly supported measure $ν$ with simple Lyapunov spectrum, we give an explicit closed-form Hölder exponent $β_*(ν, θ)$ and constant in the modulus of continuity of $λ_\pm$ in the Wasserstein-plus-Hausdorff metric, depending only on the eccentricity of $\mathrm{supp}\,ν$, the Lyapunov gap, and the Hölder index $θ$. At every $ν\in \textit{M}_c(\operatorname{GL}(2,\mathbb{R}))$ we identify the log-Hölder exponent of Tall and Viana as $κ_*(ν, θ) = θ/(2+θ)$ under a natural mixing hypothesis, and $θ/(8(1+θ))$ in the perpetuity regime. The same spectral-gap method yields a large deviation principle with explicit rate function, Hoeffding-Azuma concentration inequalities, an extension to Markov-chain driven cocycles with closed-form exponent, and a quantitative log-Hölder modulus of continuity for the integrated density of states of one-dimensional random Schrödinger operators with absolutely continuous disorder. The Hölder theory extends to $\operatorname{GL}(d,\mathbb{R})$ for the top exponent under spectral simplicity, and to the partial sums $Λ_k = λ_1 + \cdots + λ_k$ under strong $k$-irreducibility, yielding Hölder continuity of each individual sub-top exponent. A method-optimality proposition shows that $β_*$ is the best exponent obtainable from the linear balance of axioms (A1)-(A3) of the spectral-gap method; strict improvement requires either modifying these axioms or adopting a different proof strategy. A lower-bound proposition adapted from Duarte, Klein, and Santos rules out uniform Hölder continuity across $\textit{M}_c(\operatorname{GL}(2,\mathbb{R}))$.

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arXiv2026

Quantitative Analyticity for Lyapunov Exponents of Random Products of Matrices with Explicit Polydiscs and Cauchy Coefficient Bounds

The top Lyapunov exponent $λ_+(A, p)$ of a random product of matrices in $\mathrm{GL}(d, \mathbb{R})$, $d \geq 2$, with simple top spectrum, depends real-analytically on the probability weights $p$ and the matrix coefficients $A$. We establish a quantitative form of this analyticity through a single Kato perturbation argument on the complexified Markov operator on Hölder functions on projective space, yielding seven main theorems with explicit closed-form constants: (i) an explicit polydisc of holomorphy for $p \mapsto λ_+(A, p)$ in $\mathbb{C}^N$, giving the quantitative form of the Peres and Bezerra-Sánchez-Tall analyticity theorem; (ii) closed-form Cauchy bounds on its Taylor coefficients; (iii) joint analyticity in the weights $p$ and the matrix entries $A$, with explicit radii in both; (iv) an extension to Markov-chain driven cocycles, with polydisc radius explicit in the chain spectral gap; (v) explicit polynomial boundary-decay rates as $p$ approaches $\partial Δ_N$, conditional on a spectral-gap-decay hypothesis; (vi) extension to $\mathrm{GL}(d, \mathbb{R})$ for all $d \geq 2$ via the Fubini-Study metric; and (vii) a Grassmannian variant giving quantitative analyticity of the partial sums $Λ_k = λ_1 + \cdots + λ_k$ under strong $k$-irreducibility, hence of each individual sub-top Lyapunov exponent. The polydisc radius is method-optimal within the Kato class, and a Bernstein-type result shows the Cauchy growth $α! \cdot M_*/r_*^{|α|}$ is sharp up to constants. A two-matrix example with numerical values connects the bounds to the Hölder estimates of the companion paper Thiam (Nov. 2025).

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arXiv2026

Multifractal Analysis, Livšic Rigidity, and Fluctuation Theorems for Axiom A Diffeomorphisms: The Pesin Formula and the Gallavotti-Cohen Symmetry

This Part develops structural consequences of the thermodynamic formalism for Axiom A diffeomorphisms. The Pesin Entropy Formula equates the metric entropy of the SRB measure to the sum of positive Lyapunov exponents, with complete proofs of absolute continuity of conditional measures along unstable manifolds; the individual results are due to Sinai, Ruelle, Bowen, and Pesin. The Multifractal Formalism computes the Hausdorff dimension of Birkhoff average level sets via the Legendre transform of the pressure, extending earlier work of Barreira, Pesin, and Schmeling. The Livšic Theorem characterizes coboundaries through periodic orbit data with optimal Hölder regularity and an explicit norm bound in terms of the contraction rate and the Hölder exponent. The Gallavotti-Cohen Fluctuation Theorem establishes the linear symmetry relating the rate function at opposite values of the entropy production rate; for Axiom A diffeomorphisms the symmetry was established by Ruelle and by Maes, and we provide explicit bounds from the spectral gap. This Part constitutes Part VI, the final installment, of a six-part series on the thermodynamic formalism for hyperbolic dynamical systems.

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arXiv2026

Gibbs Measures on Subshifts of Finite Type: Five Equivalent Characterizations with Explicit Constants

We prove that five characterizations of Gibbs measures for Hölder potentials on topologically mixing subshifts of finite type are equivalent: the Jacobian condition, the classical cylinder-based Gibbs property, the eigenmeasure of the Ruelle transfer operator, the variational equilibrium state, and the minimizer of the large deviations rate function. The equivalence is established in a single theorem with explicit constants expressed in terms of the Hölder exponent, the potential norm, the alphabet size, and the mixing time. The proof yields explicit spectral gap estimates for the transfer operator via the Birkhoff cone contraction technique, Lipschitz stability of the Gibbs measure in Wasserstein distance under perturbation of the potential, and statistical limit theorems including a central limit theorem with Berry-Esseen bounds and a large deviations principle. This Part constitutes Part I of a six-part series on the thermodynamic formalism for hyperbolic dynamical systems.

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arXiv2026

The Convex-Analytic Structure of Thermodynamic Equilibrium: Pressure, Subdifferentials, and Phase Transitions

We develop the convex-analytic structure of the thermodynamic formalism for continuous maps on compact metric spaces. The pressure functional is the Legendre-Fenchel transform of the negative entropy, and the biconjugate recovery of the entropy from the pressure establishes a complete duality. Equilibrium states are elements of the subdifferential of the pressure, uniqueness of equilibrium states corresponds to Gâteaux differentiability, and first-order phase transitions correspond to non-differentiability. For systems with specification and Hölder potentials, the pressure is Fréchet differentiable in the Hölder norm, and the second derivative of the pressure equals the asymptotic variance of the Birkhoff sums. We prove a universal variational principle that unifies the classical additive, the subadditive, and the relative variational principles through a single theorem on convex functionals satisfying convexity, lower semi-continuity, coercivity, and cocycle invariance. Extensions to systems with the specification property and to non-compact spaces under coercivity conditions are included, with applications to countable Markov shifts via Sarig's recurrence classification. This Part constitutes Part II of a six-part series on the thermodynamic formalism for hyperbolic dynamical systems.

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arXiv2026

Uniform Hyperbolicity and Symbolic Dynamics: Markov Partitions, Shadowing, and the Coding of Axiom A Diffeomorphisms

This Part establishes the geometric theory of uniformly hyperbolic sets with explicit quantitative bounds throughout, and contains five main theorems. The Stable Manifold Theorem is proved via the backward graph transform, with a complete fiber-contraction argument yielding $C^r$ regularity and Hölder dependence of the local stable and unstable manifolds on the base point, with explicit manifold-size estimate in terms of the contraction rate $λ$ and the second-derivative bound of the diffeomorphism. The Spectral Decomposition Theorem gives the unique decomposition of the nonwandering set into basic sets, with explicit mixing rates for the topologically mixing factors. The Shadowing Lemma provides explicit error bounds controlling how far a pseudo-orbit deviates from a tracking true orbit. The existence of Markov partitions of arbitrarily small diameter is established constructively, with explicit diameter bounds expressed in terms of the shadowing constants. Finally, the coding map from the subshift of finite type to the hyperbolic set is constructed and shown to be Hölder continuous with quantitative control on the exceptional set where it fails to be injective. Along the way we establish canonical coordinates through the bracket map with quantitative bounds. All constants are expressed in terms of the contraction rate, the Hölder exponent of the derivative, the manifold dimension, and the injectivity radius, providing the quantitative infrastructure required to transfer the symbolic spectral theory of Part I Thiam (2026a) and the variational theory of Part II Thiam (2026b) to the smooth setting. This Part constitutes Part III of a six-part series on the thermodynamic formalism for hyperbolic dynamical systems.

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