For polynomially mixing billiards with cusps, Birkhoff sums of observables φ(x) = d(x,x0)^{-2/α} with tail index α satisfy stable laws whose index is a function of both α and the mixing exponent γ when γ ∈ (1/2,1) and α ∈ (0,2) excluding 1.
Invariant measures and dynamical systems in one dimension
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Explicit construction of invariant u and scaled v for the transfer operator of Parry-type β-expansions gives sharp L^∞ asymptotics P^k F = u + β^{-k}(F(1)-F(0))v + o(β^{-k}) for unit-integral smooth F.
For Parry-type beta-expansions, the Perron-Frobenius operator has eigenvalue 1 attracting Lipschitz functions exponentially in L1, while its point spectrum on L^p (1≤p≤2) contains the entire open unit disk.
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Stable laws for heavy-tailed observables on polynomially mixing billiards
For polynomially mixing billiards with cusps, Birkhoff sums of observables φ(x) = d(x,x0)^{-2/α} with tail index α satisfy stable laws whose index is a function of both α and the mixing exponent γ when γ ∈ (1/2,1) and α ∈ (0,2) excluding 1.
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Sharp iteration asymptotics for transfer operators induced by greedy $\beta$-expansions
Explicit construction of invariant u and scaled v for the transfer operator of Parry-type β-expansions gives sharp L^∞ asymptotics P^k F = u + β^{-k}(F(1)-F(0))v + o(β^{-k}) for unit-integral smooth F.
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Spectral and dynamical results related to certain non-integer base expansions on the unit interval
For Parry-type beta-expansions, the Perron-Frobenius operator has eigenvalue 1 attracting Lipschitz functions exponentially in L1, while its point spectrum on L^p (1≤p≤2) contains the entire open unit disk.