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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2502.17113 |
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| _version_ | 1866918510521745408 |
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| author | Cornean, Horia D. Sørensen, Kasper S. |
| author_facet | Cornean, Horia D. Sørensen, Kasper S. |
| contents | We consider base-$β$ expansions of Parry's type, where $a_0 \geq a_1 \geq 1$ are integers and $a_0<β<a_0+1$ is the positive solution to $β^2 = a_0β+ a_1$ (the golden ratio corresponds to $a_0=a_1=1$). The map $x\mapsto βx-\lfloor βx\rfloor$ induces a discrete dynamical system on the interval $[0,1)$ and we study its associated transfer (Perron-Frobenius) operator $\mathscr{P}$. Our main result can be roughly summarized as follows: we explicitly construct two piecewise affine functions $u$ and $v$ with $\mathscr{P}u=u$ and $\mathscr{P}v=β^{-1} v$ such that for every sufficiently smooth $F$ which is supported in $[0,1]$ and satisfies $\int_0^1 F \; \mathrm{d} x=1$, we have $\mathscr{P}^kF= u +β^{-k}\big ( F(1)-F(0)\big )v +o(β^{-k})$ in $L^\infty$. This is also compared with the case of integer bases, where more refined asymptotic formulas are possible. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_17113 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Sharp iteration asymptotics for transfer operators induced by greedy $β$-expansions Cornean, Horia D. Sørensen, Kasper S. Dynamical Systems 37 (Primary) 39, 40, 47 (Secondary) We consider base-$β$ expansions of Parry's type, where $a_0 \geq a_1 \geq 1$ are integers and $a_0<β<a_0+1$ is the positive solution to $β^2 = a_0β+ a_1$ (the golden ratio corresponds to $a_0=a_1=1$). The map $x\mapsto βx-\lfloor βx\rfloor$ induces a discrete dynamical system on the interval $[0,1)$ and we study its associated transfer (Perron-Frobenius) operator $\mathscr{P}$. Our main result can be roughly summarized as follows: we explicitly construct two piecewise affine functions $u$ and $v$ with $\mathscr{P}u=u$ and $\mathscr{P}v=β^{-1} v$ such that for every sufficiently smooth $F$ which is supported in $[0,1]$ and satisfies $\int_0^1 F \; \mathrm{d} x=1$, we have $\mathscr{P}^kF= u +β^{-k}\big ( F(1)-F(0)\big )v +o(β^{-k})$ in $L^\infty$. This is also compared with the case of integer bases, where more refined asymptotic formulas are possible. |
| title | Sharp iteration asymptotics for transfer operators induced by greedy $β$-expansions |
| topic | Dynamical Systems 37 (Primary) 39, 40, 47 (Secondary) |
| url | https://arxiv.org/abs/2502.17113 |