Invariant distributions of partially hyperbolic systems: fractal graphs, excessive regularity, and rigidity
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| Format: | Preprint |
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2024
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| _version_ | 1866908257747992576 |
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| author | Xu, Disheng Zhang, Jiesong |
| author_facet | Xu, Disheng Zhang, Jiesong |
| contents | We introduce a novel approach linking fractal geometry to partially hyperbolic dynamics, revealing several new phenomena related to regularity jumps and rigidity. One key result demonstrates a sharp phase transition for partially hyperbolic diffeomorphisms $f \in \mathrm{Diff}^\infty_{\mathrm{vol}}(\mathbb{T}^3)$ with a contracting center direction: $f$ is $C^\infty$-rigid if and only if both $E^s$ and $E^c$ exhibit Hölder exponents exceeding the expected threshold. Specifically, we prove:
If the Hölder exponent of $E^s$ exceeds the expected value, then $E^s$ is $C^{1+}$ and $E^u \oplus E^s$ is jointly integrable.
If the Hölder exponent of $E^c$ exceeds the expected value, then $W^c$ forms a $C^{1+}$ foliation.
If $E^s$ (or $E^c$) does not exhibit excessive Hölder regularity, it must have a fractal graph.
These and related results originate from a general non-fractal invariance principle: for a skew product $F$ over a partially hyperbolic system $f$, if $F$ expands fibers more weakly than $f$ along $W^u_f$ in the base, then for any $F$-invariant section, if $Φ$ has no a fractal graph, then it is smooth along $W^u_f$ and holonomy-invariant.
Motivated by these findings, we propose a new conjecture on the stable fractal or stable smooth behavior of invariant distributions in typical partially hyperbolic diffeomorphisms. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2411_19665 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Invariant distributions of partially hyperbolic systems: fractal graphs, excessive regularity, and rigidity Xu, Disheng Zhang, Jiesong Dynamical Systems We introduce a novel approach linking fractal geometry to partially hyperbolic dynamics, revealing several new phenomena related to regularity jumps and rigidity. One key result demonstrates a sharp phase transition for partially hyperbolic diffeomorphisms $f \in \mathrm{Diff}^\infty_{\mathrm{vol}}(\mathbb{T}^3)$ with a contracting center direction: $f$ is $C^\infty$-rigid if and only if both $E^s$ and $E^c$ exhibit Hölder exponents exceeding the expected threshold. Specifically, we prove: If the Hölder exponent of $E^s$ exceeds the expected value, then $E^s$ is $C^{1+}$ and $E^u \oplus E^s$ is jointly integrable. If the Hölder exponent of $E^c$ exceeds the expected value, then $W^c$ forms a $C^{1+}$ foliation. If $E^s$ (or $E^c$) does not exhibit excessive Hölder regularity, it must have a fractal graph. These and related results originate from a general non-fractal invariance principle: for a skew product $F$ over a partially hyperbolic system $f$, if $F$ expands fibers more weakly than $f$ along $W^u_f$ in the base, then for any $F$-invariant section, if $Φ$ has no a fractal graph, then it is smooth along $W^u_f$ and holonomy-invariant. Motivated by these findings, we propose a new conjecture on the stable fractal or stable smooth behavior of invariant distributions in typical partially hyperbolic diffeomorphisms. |
| title | Invariant distributions of partially hyperbolic systems: fractal graphs, excessive regularity, and rigidity |
| topic | Dynamical Systems |
| url | https://arxiv.org/abs/2411.19665 |